This is a guide for new players of Dragon Coins who do not plan to spend much, or possibly any, money playing the game. The play experience for a free player is likely to be fundamentally different from that of someone spending money to get strong monsters early on, and this guide is intended to familiarize free players with the mechanics of the game and illustrate how they may best make progress in the game and develop a strong collection of monsters. Hopefully some of this will also be useful for those who do open their wallets to support the game and advance their progress, but a lot of the advice below is specific to those who will feel the restrictions of free play.
Starting Out
Any time you're starting Dragon Coins for the first time on a device, you have to go through the tutorial and create a new account - even if you're moving from one device to another and want to enter a game ID and password that you've created elsewhere. The tutorial covers some useful info, glosses over a lot of details, and at the end of it you're level four, with a small group of common monsters, a U red dragon, an R monster you've picked to be your first leader, and one of a few possible R monsters from the introduction to picking from the summoning chests. A reasonable question is to ask, "Now, what?"
(Footnote to the tutorial: it doesn't really matter which first leader you pick; they all have the same skill in battle, and comparable stats. Marina and it's evolution Vepar have a slight edge on attack, Imp/Ifreet on HP, and Ponicorn/Unicorn are in the middle, but the differences are very minor. Personally I found I had more Fiends than Beasts or Celestials in the early going, which would have made Vepar the best in terms of its leader skill (but I'd picked Ponicorn), but since all of them are available as rewards in story mode and frequently in the Friend and Rainbow chests, you can eventually collect them all. The wood-type Ponicorn is at a disadvantage in the first story missions against the Fire monsters, but the missions are so easy that you'll cruise through them anyway if that's your pick.)
Initially you're going to concentrate exclusively on Story Mode, and the first chapter's areas will reward you with monsters that are strong against each upcoming area. Each mission has gem rewards that give you new monsters, most of which you'll fuse to other monsters you have in order to make them stronger. If you collect all the different gems available in a mission, you'll get a one time only bonus gem containing, generally, a slightly stronger monster. In the first chapter of the story mode, you only get commons as bonus gems, except for an uncommon Brickle at the end of the chapter. Getting every gem from every mission in an area will give you a 50 rainbow coin reward - small, but it adds up over time.
As you progress, you'll find that you quickly fill up your monster box, which leads to a question of how to choose what monsters are worth levelling up with fusion.
Fusion, and what is worth levelling up
Commons are not worth levelling up, period. No early mission will require you to have done so, and once you get a little bit into the game you will quickly be able to replace them with stronger monsters, even if you max out and evolve them to Common+. The only long-term reason to level and evolve common monsters is to fill out entries in your monster tome, and if you really want to do that, you can do it later when you can afford to blow fusion fodder on monsters that you'll never use after your first couple hours with the game.
Many uncommon monsters are also terrible, although there are some that are surprisingly good which can be useful for you for some time to come. My general rule for a monster that's worth keeping in the early game is that you should expect its attack/charge to get close to 200 by level 20. Levelling a monster to 20 is pretty fast, and it's not hard to get all the monsters you regularly use to at least that point. It's unfortunately not that easy to work out if this will be the case for a monster by looking at its wiki entry, but generally if its max attack/charge is over 300, you can expect to be doing pretty well by level 20.
(Footnote on unusually strong uncommon monsters. In terms of actually growing up to have respectable attack values, I like Zombling/Zombie from Shadow, Knitebrite/Ghost from Light, and the null Brickle/Golem. The uncommons from each color with an element "Surge" ability may be worth using early on, to boost damage against strong monsters, but are not good long-term monsters; of those, the best is probably the blue Bladefin/Megaladon - if you do evolve it, it benefits from its own Beast Fire leader ability, and if you have other beasts, it might be worthwhile. The light Cupidette/Angel is one of few early monsters with a strong healing ability, and can be worth levelling and keeping on a team for that reason.)
Initially, you will probably see the best results from funnelling most of the monsters you collect into your starting leader and the R monster you got from starting out. It's possible that you'll get some good or great monsters from the Friend Chest or from Rainbow or Feature tickets you get from daily rewards. In general it is worth trying to stick exclusively to fusing fodder monsters to a monster of the same color, since you get a much better return in terms of experience. Early on, it is entirely possible that you won't have an appropriate monster to target with certain colors - if so, it's not wrong to fuse them to one of your starting rares. Try not to do this more than you have to, though; even though commons sell for very low amounts of gold (50 and up), it may be better to sell for pocket change than to spend larger amounts of gold to fuse for small results.
Some Null-element monsters will appear to be useless, with terrible attack and hp values, and they generally fall into one of two categories. Gray Worms, Iron Butterflies, and Fuzilites of various sorts are worth an unusually high amount of fusion experience, and should be fused to monsters you want to level up and use for a long time. Skyworm/Scarab, Midasbane/Grootslang, and Bonus Beast/Mammon are simply worth a large amount of gold when you sell them. There is no better use for these latter monsters than selling them! In particular, do not make the mistake that I've seen some people make of getting an R Bonus Beast or Magic Fuzilite and using it on your team or setting it as your leader because you think all R monsters are good! If you set one of these monsters as your leader, it will signal to other players that you don't know what you're doing, and they will not want to be your friend. Sadly, you are not able to send someone a message to tell them, "Please pick literally any other monster as your leader," no matter how much you may want to.
Friends and their uses
Friends are other players whose leader monsters will show up among the available allies whenever you start a new mission. Each friend's leader is available once per day (it resets at midnight, Pacific time zone). Choosing a friend's leader as an ally gives both players 10 friend coins (which can be used at the Friendship Chest to get new monsters, which is an important, if extremely random, method of improving your teams), and if that monster has been evolved to have a + by its rarity, it will also give a leader skill which may or may not be particularly useful, but is never bad. Choosing a non-friend's monster will only give 5 coins to each player, and thus it's generally better to choose friends for allies. However, the way you make friends is by using a non-friend's monster to complete a mission, so whenever you see someone with an ally monster you would like to be able to use regularly, it's worth using them for a mission and seeing if you can try to get a new friend. Having friends with strong monsters as their leaders can make a big difference in being able to successfully complete tough missions.
It's a simple fact of Dragon Coins that players stop playing it all the time. Particularly, new players often play actively for a short period and then find other things to do with their time. Thus, since one of the primary benefits of having friends is that they may choose your leader for a mission and thus give you friend coins, it's a good idea to periodically go through your friend list and prune away inactive players. If a player has been inactive for more than a week, there's a strong chance that you will never see them log in again, and if they do, not often enough to be a reliable source of friend coins. Unless they have a very strong leader that you enjoy having once/day access to, it's often better to unfriend that player to free up a slot for someone more active. It can take a while to fill up your friend list initially, but eventually you will max out at 50 friends, and when you do it is valuable to maximize how much those friends are helping you. It might seem a bit callous to prune away "friends", but frankly, these are probably not people you know, and you are unlikely to get to know them unless both of you participate in online discussion; plus, if they stop playing the game, they're not going to notice, anyway.
Unless you participate in the Dragon Coins forum on Sega's site, you are extremely limited in how you can communicate with your friends. You can send a mass message to all your friends once per hour, and once per day doing so will give you 200 friend coins. Since players are therefore encouraged to spam their friends whether or not they have anything to communicate, most communications from other players should be assumed to be effectively meaningless, particularly the request to switch to a fire-type leader, since that's the first message on the list of available communications.
The monster that shows up as your leader to your friends is the leader monster for your main team. This may or may not be a monster you want to take to every mission, and if it's not, you should set up one or more subteams with groups of monsters you would prefer to use.
Team Selection
You can make a main team, as well as up to 9 sub-teams from which you can choose before each mission. Again, the leader of your main team will be the monster available to other players to choose as their ally - it is important to make sure that monster is one which is attractive to other players. It doesn't have to be a monster with leader skill, so long as it's the strongest that you have.
In the first chapter, all monster in a mission will be the same element, and so it is often best to bring a team comprised of monsters that are all strong against that element - fire vs. wood, wood vs. water, water vs. fire, shadow vs. light, light vs. shadow. That said, if you have a strong monster such as your starting leader which is not weak against that element, it may be worth bringing it along anyway. It's worth sorting your monster box by Attack, which specifically sorts by attack/charge - a monster that gets 200 attack for one charge is stronger than one that hits at 300 for two charges. For the most part, early on, bring the four strongest monsters you have of the right element; if you only have a couple strong monsters in that element, add in other strong monsters you have as long as they are not weak to the mission's monsters.
Later on, missions will have combinations of elements. Sometimes, when you are about to start a mission, it will show "rules" that change the strengths of monsters you bring, under certain conditions. Pay attention to those rules, when they show up - it is often worth making a team that fits those rules, even if it's not the strongest monsters you could bring, in return for the bonuses your team will get. If there's no guidance, pay attention to the color of the mission when it is displayed in your story mode list. Generally, at the very least, you don't want to bring monsters weak to wood if the mission listing is green. That said, the game is sneaky, and often if a mission is shown as red, for example, you might see wood monsters as well that will be strong against any water monsters you bring to put out the fire. If you fail a mission, it's usually possible to make a new team that can beat it, although you may find that success eventually becomes heavily dependent on having a strong ally.
Evolution
Generally speaking, if you level a monster to 20 and have a spare copy of that monster, you can evolve it to its "+" form (for exceptions, see the Dragon's Road and Urgent Missions sections). Evolving a monster gives it higher stats across the board, increases its maximum level, and also gives it a leader skill which may or may not be particularly useful. Leader skills are often ones which buff monsters of a particular race (Beasts, Fiends, etc), and you probably won't be able to build a strong team of all or mostly one leader type. Early on it's likely that leader skills you have access to will not be particularly strong, though there are a few game-changingly powerful ones you may get access to in the long run.
One of Lily's quests for new players is to level a rare to 20, at which point she will gift you with another copy of that rare with which you can evolve the first monster to R+. If you get a strong rare early on, it's possible that it may be worth fusing it to 20 before you do the same with your starting leader, especially since it's harder to get a duplicate of most other rares compared to the starter leaders (which show up as story rewards as well as in chests). Take a look at the wiki to see what the stats and leader ability are like on the evolved forms of other rares you get, and see if one is worth making that effort for. Otherwise, you can do a lot worse than levelling up your starting leader, evolving it, and continuing to level it. You might find it easier to get other players to accept your friend requests if your leader is a strong monster that's not common among the active players around your level, just because most players' friend lists will be full of people using the starter monsters.
Not all monsters are worth evolving, even if you get a second copy of it! Again, check the wiki to see how strong a monster can become, and if the answer is, "Actually, not that strong," then it is OK to sell (or occasionally fuse) that monster away.
Your monster collection: what to keep, and what to trash
Every common and nearly every uncommon monster is ultimately expendable. There will be uses later on for certain specific uncommons (see the entry on the Urgent Missions and the Hero chest), but when that time comes you will be able to get what you need, generally. For the most part, unless you know for sure that you will need an uncommon monster soon, it's better to fuse or sell them than have them just use up space in your monster box.
Not all rares are good. Not even all Super Rares are good. A free player will get a small number of rainbow and feature chest tickets from daily bonuses, and will eventually get Friend Coins you can use to draw from the Friendship Chest, and there's a small chance of getting a super rare from that chest (I didn't for more than 100 days of playing, but then I got two within a week of each other). If you get a super rare, you'll probably be excited even if it's a terrible one like Coco or Festa. Don't be led astray! Don't set a super rare as your leader unless it's one that you would be excited to see being used by one of your friends. Otherwise, a boring but useful rare is a better choice. Some super rares are so bad that they are not even worth letting them take of space in your monster box. If you don't have any other SR monsters, you probably won't be able to bring yourself to sell them, but at least don't sink your limited resources into fusing them up to high levels when the reward will not be worth what you spend.
Some monsters are not useful for missions but may have good skills to use in the Arena (see that section for more details); in particular, arena skills like Poison Blind, Go Big Or Go Home, Skill Seal, Floppy Pusher, Ejector, and Tiny Silver may merit keeping a monster around even if you won't use it for other purposes.
If you follow the Dragon's Road (see below), you'll find a lot of your monster box taken up with dragons and have to make some hard choices about what other monsters to keep; generally, stick to stuff you want to use, and if monsters aren't good enough to take along for missions, they're not worth keeping. Sometimes take a look to see if a comparatively weak monster has a potentially useful leader skill if you get a chance to evolve it - that can make it worth holding on to (for example, Summer Skeeter has fair at best stats, but Summer Lilith's "Fiend Shower" ability is awesome on the right team; it also has a good arena skill).
Rainbow coins, and their uses
Dragon Coins has a lot of different kinds of coins, acquired in a variety of ways, and the most elusive for a free player are Rainbow Coins. Primarily acquired by spending money in the Shop, but also available 50 at a time for getting 100% of the coins in a story area, occasionally as daily bonuses in first few weeks for new players, and as event rewards. If you are starting from scratch and not spending money, you should assume that you will not see a lot of these, and so you need to be careful how you spend them. The most enticing way to use them is for chest pulls from the Feature Chest - if you save up 1500 of them, you can 5 monsters of R or better, including a guaranteed SR or better monster. Most SRs are good, and some are transformative. However, some SRs are abjectly awful, and many are no better than a good R; unless you start placing in events and getting feature tickets, you won't be able to manage enough pulls from the feature chest to be able to bank on having some really strong monsters, and plenty of players can tell tales of blowing all their rainbow coins and getting practically nothing good for them. My recommendation, if you are in the game for the long-term, is not to spend your coins on chest pulls. Certainly, don't spend rainbow coins on the Rainbow Chest, which has poor odds of yielding SRs and can give you monsters as weak as U+, which (per above discussion) are not worth your time.
Then, the shop offers a few ways of getting short-term disposable bonuses; Stamina Boosts to let you keep playing after you use all your Stamina; Liquid Courage to strike at a boss or arena after you've exhausted yourself, and Revival Medicine for trying again on a mission that has beaten you. Don't get any of these, but especially do not spend rainbow coins on revives. You can always wait to regain stamina and try a mission again, but rainbow coins are not easily regained without spending money.
Lastly are Monster Slots, available 5 at a time for 150 coins. This is, I've come to feel, essentially the only reliably useful way to spend rainbow coins. It's the only thing you can buy which can't later be lost or become useless. Your monster box will quickly became very cramped, particularly if you want to evolve dragons (again, see below), and without spending coins your box will max out at less than 50 slots after you've had all the slots that come from daily rewards. Spending 600-900 coins on 20-30 slots is a very good investment in your future; few things in the game are more frustrating than maxing out your box and not wanting to get rid of any monsters. You can always get more stamina and courage with patience, and eventually your teams will get strong enough that you can earn some chest tickets in events to expand with more interesting monsters, but if you don't have someplace to put them, you'll dead-end in the game.
Gold management
Gold coins are tricky - you need them for fusion and evolution costs, but most missions don't have high gold rewards, and even if you are inclined to spend money on the game, you can't directly buy gold, or even buy rainbow coins and turn them into gold. It's a very peculiar omission, to my mind, and it means that if you run short on gold there are few really palatable options. Among those, you can complete missions and sell the monsters you get instead of fusing them. Commons in particular are not worth a lot of gold (50 each at level 1), but it adds up over time, and sometimes it may be the only way you can afford a steep evolution cost. Still, it can be disheartening to work on missions only to throw away the monster rewards you get, so it's preferable to find other options.
If you don't mind linking your social media accounts to the game, you can do so to enable the game to post on Twitter/Facebook when you draw rare or better monsters. Assuming that you have social media accounts that you don't mind having become full of Dragon Coins spam, you get 1000 gold for each post, meaning that drawing a rare can get you 2000 gold, plus more if it's a duplicate that you can sell. Also, if you get a rare you definitely don't want, they sell for a minimum of 2000 gold. There are some that sell for far more - there are palette swaps of the starting leaders (Embercorn, Sylvina, and Frost Imp) that can be sold for 8000 gold each, though it's worth noting that they are also worth extremely large amounts of fusion experience.
(Footnote regarding getting gold from social media posts: if you draw more than one of the same rare within a short period of time, which is very common when doing a lot of Hero Chest draws, Twitter and Facebook will reject posting exactly the same tweet/status within a short period of time; you can tap the box that shows what the game is going to post and add a random word or something to the end before you post it to keep your post from being rejected. If the post fails, that's often what the problem is, and you should try to avoid that since you don't get gold for failed posts.)
Good fusion practice will help you conserve your gold, as well. Never fuse unless you can fuse 5 monsters at once, or to get as close to level 20 as possible for a monster you want to evolve but not further train (usually a dragon). Always try to use fusion fodder of the same color as the monster you are levelling up, except if it's a null like a fuzilite. If you're levelling a monster past level 30 or so, avoid using level 1 commons as much as possible - they give a weak reward and the cost is the same if you are using stronger monsters as fodder. If you're levelling up SR monsters, save any grayworms or fuzilites for them - don't use them to boost up a new level one monster whose fusion costs are low.
Dragon's Road: it's paved with lots and lots of dragons
In the absence of reliably drawing from any chest besides the Friendship Chest, the best way to get strong monsters is by evolving dragons. Dragons are unusual in that each rarity of dragon can be evolved beyond the + form; C+ dragons can evolve to U, U+ to R, and so on all the way to SR+, and the SR+ dragons are among the strongest in the game, particularly in terms of raw attack and health stats. It's a real long road to get there, though.
That road comes in the form of daily missions at three difficulty levels, with cost of 3, 6, or 12 stamina. Monday through Friday, the different colors are represented, and each day you can get rewards by collecting gems that give you dragons that can be levelled up and evolved. The 3 and 6 stamina cost missions have a reward of a U rarity dragon, while you can get a U+ from the hard mission. The gem rewards for each mission will all be monsters of a single color, so you can grind those missions to level up the dragons you get. Further, the dragon's road missions generally give higher rewards per stamina invested than story missions. In particular, the 6-cost mission will give you enough gold to cover your fusion costs, while the 3-stamina mission will run a little lower than that.
It's a really long road to go from C to SR+, though once you have strong enough teams that you can get U+ monsters more easily, it's not that hard to get R dragons, which are all pretty strong. Light and Shadow R and R+ monsters in particular have a really strong ability called Mounting Pressure which lets you reset all enemy monsters' attack counts to 25, which is one of the best abilities you can get as a newish player. It's the kind of thing that makes it much easier to do harder missions, so evolving a couple of those should be a high priority, once you can do the dragon's road missions. The primary color dragons, on the other hand, are mostly valuable for having high attack values. In general, I found that having a team of monsters strong against the daily element and with an average attack/charge of 200 or better should mean that you can beat the 3 and 6 cost missions, and you don't need too much more than that to be able to squeak out the hard mission.
Evolving dragons is a little weird compared to most other monsters; where mostly you just need a level 20 monster and then another copy of the same monster, for any given rarity of dragon, the evolution requires another copy of that dragon, as well as another one of the same rarity from a different color. So, Light Dragons will also require a shadow, red for shadow, blue for red, green for blue, and light for green. Thus, you won't get in a single day everything you need for an evolution, but if you get the daily rewards every day for a week you should be able to make a couple rare dragons.
Weekends, generally, offer the Dragon's Trove, with high experience and gold rewards, as well as valuable gold and fusion monsters as daily rewards. The hard Dragon's Trove mission, in particular, rewards light and shadow U+ dragons for completion, which is fantastic. Once you can do that mission reliably, grinding it on the weekends will really speed you along the dragon's road. There was a period of a couple months in Summer 2014 where the Trove was replaced by Joker missions, but that seems so far to have been a one-time experiment.
Urgent Missions, the Hero Chest, and building your own SR+ monsters from spare parts
Besides the Dragon's Road, there's one other sure way to eventually get SR+ monsters. At the time that I'm writing this, every Friday through Monday is an event where each mission you complete has a chance to offer you an Urgent Mission after you complete it (assuming that you are at least level 10). These urgent missions are all one-round fights against an SR+ boss monster. The missions start easy, but each one you get after the first will increase in level, until eventually they may be too strong to kill by yourself. Whenever you complete one of these missions, you'll get a reward in Hero Coins that can be spent a hero chest that contain, among other things, some rare monsters not available anywhere else.
Typically (Joker is an exception) there are 4-5 of these rares, and each one, to evolve to R+, requires a U+ dragon of the appropriate color, along with a different U+ uncommon (the U version of which will also be found in the hero chest). If you can get one of each rare, along with the appropriate uncommons, you can evolve them all to R+, and then all of them together can be evolved to an SR monster. Do that twice, and the two SRs can become an SR+. It's pretty unlikely that you'll be able to get everything you need during one event, but the events cycle periodically, so currently every 5 weeks you'll see each of the Urgent Mission bosses once each.
Some of these monsters are a lot stronger than others. Of them, Konohana Sakuya (green) with strong basic stats and the Possess ability is by far the most useful to a player without a wide range of monsters. Pirate Queen Alwida (blue) has a strong attack and an ability which makes 2-charge monsters stronger; she's definitely useful on the right team. Joker (shadow) has an ability (Share the Wealth) which is fantastic and otherwise only available on SR/SR+ monsters, but her stats are very poor otherwise. Ahriman (red) has great health, tolerable attack, and an ability of only moderate use, while Koryu has very high health, poor attack, and an ability you probably won't find very useful. If you have to choose which monsters components to save in your limited monster box slots while you wait for their event to come around again, prioritize Konohana and Alwida.
Some of the component R and R+ monsters are actually quite strong and worth holding onto in and of themselves, though the bulk of them are pretty weak. The weaker ones are still better than stock U/U+ monsters, but you should be able to replace most of them with better monsters before long. Here are ones I think might merit a place on your teams for a longer time:
Ahriman: Spirit Right Hand/Taromaiti (blue), Spirit Heart/Aka Manah (Light)
Alwida: Polly Phonic/Hua Po (green), Fire Dancer/Etoile (red), Ghost Ship/Flying Dutchman (light)
Koryu: All but Suzaku (red) are actually pretty good, but especially Byakko Orb/Byakko (Shadow)
Joker: Jack of Clubs (green), King of Diamonds (light)
Konohana: all have strong abilities (for single color teams) and poor stats; none are great, but Cottontail/Vorpal Bunny is arguably the least bad, and has a strong Arena ability
Arena
Starting, I believe, at level 10, players can place their monsters in an Arena which other players can attack during Arena Events (at a cost of one courage for each attack). Like monsters in tough story missions, enemies in the arena use skills which can make them a real challenge to defeat, if they are chosen well and arrayed intelligently. If your arena defeats your opponent, or if you successfully attack someone else's arena, you earn Battle Coins which can be used to draw from the Battle Chest, which contains monsters considered to be strong in the arena (sometimes this isn't really the case), up to SR rarity. At the time of this writing, this may actually be the best way for a free player to get SR monsters. Arena strategy deserves its own guide, and for a player with a limited monster selection and limited box space, you may not want to keep monsters around just for strong arena abilities, but the rewards can be substantial, so it may be worth investing time into. I was already high level when the Arena premiered, so I can't really speak to what it's like for a low level player, but it's probably worth participating in to get the battle coins and a chance at some (mostly) strong SRs. Also, it's good practice to learn how to beat enemies that use skills.
Generally, the strongest Arena abilities for monsters are Go Big or Go Home, Mini Silver, Skill Seal, Floppy Pusher, Ejector, Pressure, and Poison Blind. Confusion Bomb will get some random wins if it falls the right way, but players generally learn to guide it to their weaker monsters or over the edge, and it will only rarely stop a strong attacking team. A couple abilities have a tendency to work against the defender, particularly Escalate and Trouble Towers, and those should be used only with caution, if at all.
Tactics of play
Fundamentally, each mission you begin has the same goal: guide coins to your monsters so that you can exchange them for damage dealt by your monsters to the enemies until every enemy is dead and you collect the gem rewards. Note that there's nothing there about keeping all your health. An important thing to remember is that it can be worth deliberately taking damage to reset enemy attack counts and be able to drop more coins, especially if there are heal coins that you are close to being able to collect.
Likewise, the bonus from Lily's comeback cube is something you never get if you never drop to critical health; sometimes it can actually be worth deliberately falling to low health in order to get the cube bonus - because of randomness, this is unreliable, but worth keeping in mind. Sometimes getting the right comeback cube makes the difference between success and failure on a tough mission.
Initially, most players seem to play Dragon Coins very deliberately, making the most of each coin placement. There's nothing wrong with that, but on easy missions you can often succeed by just dropping lots of coins very quickly, letting enemy attacks trigger, and healing from the coins they drop nearly as fast, or faster, than they can hurt you. Also, keep in mind that Dragon Coins supports multi-touch - if you tap in two or more locations on the screen at once, you'll drop coins simultaneously in each place. If you have an ability that protects you from enemies, like Parry, or if you get to use the rare Stop ability that keeps enemy attack counts from going down at all, you can just go nuts with 4-5 fingers at once, dropping coins and watch enemy health melt away.
Actually spending money on the game
As discussed above in the Rainbow Coins section, monetization of Dragon Coins is kind of weird. It's hard to get a certain, lasting result from spending a moderate (say, $5-25) amount of money, and even spending $100 or more may not give you all that you want. At the time that I write this, there's a starter pack available for $5 that gives a little of all the things you can buy with money in the game, and it's not a bad value. You could get that and never spend anything else on the game, if you like. If you find that you like the game but have a hard time due to limited monster box space, it's very reasonable to think about buying some rainbow coins to expand your monster box. Again, box slots are the only thing you buy that can never go away, and as you collect more monsters and focus on complex evolutions, they will often be the biggest limiting factor in how easily you can pursue your goals.
Long-term goals
Every player who gets hooked on Dragon Coins will eventually lose interest and find other ways to occupy their time. If you want to maximize the enjoyment you have in the game once the initial thrill of seeing all the shiny coins fades away, you'll probably need to set some goals that you can work towards over time, letting the fulfillment of progressing toward them help keep you engaged with the game. These can include story progress (though the story mode is likely to get hard faster than your monsters become strong, at least for a while), evolving dragons, and evolving Urgent Mission bosses.
It is a bad idea to set a goal of collecting every monster. Most monsters cannot be obtained without spending rainbow coins or chest tickets, and even if you spend a lot of money on coins, it's really hard to get a complete collection. One player famously posted on the forums about spending more than US$5000 on rainbow coins, and still falling well short of collecting all the monsters. Ultra Rares by themselves could break a person's spirits, but even getting all the SRs is well out of reach for the vast majority of players. It's good to appreciate getting new monsters, but this is not a game conducive to long term satisfaction if you gotta catch 'em all (sorry, wrong game).
Eventually it's possible for free players to get to the point that you can do surprisingly well in the ranking events that are currently always going on. Besides the hero and battle coins you can earn, it's not that hard to get to a point where the event rewards include rainbow or feature chest tickets for free chest draws. At the time of writing, I've had more than 30 feature chest draws despite never spending rainbow coins to do so, and I've amassed a modest set of SR and SR+ monsters that are strong enough to allow me to beat nearly any urgent mission boss and feel like I have a shot against even the toughest story missions. It took a lot of playing and a long time to get to that point, but it's achievable.
Dragon Coins can be bizarrely addictive. If you feel compelled to play the game but aren't having fun, it's a good idea to step away from it for a little while and think about why you are playing it, and to what degree it is giving you pleasure. If you're having fun with it, great! If not, I generally advise against playing games which are not fun for you. Take care of yourself, first and foremost, and don't stress out over games if you can avoid it.
Conclusion
I really hope some people find this guide helpful. This game, though it drives me crazy in a variety of ways, has gotten itself into my head in a way that very few others have, and I know that's true for a lot of people. I wish everyone well who plays it, and I hope that those who can't or choose not to throw much money at it can find ways to enjoy and have fun with it.