This question is akin to asking "I'm going to the Western hemisphere, what should I eat?". Watch "A bite of China" (the first season is on youtube) to get some tips on the variety and styles of cooking available. In general you'll want to specifically avoid anything you might be familiar with (e.g. fried rice, fried noodles, sweet & sour etc. - that's if you can find them!) as you'll just be disappointed because you'll probably be expecting a better version of what you've had before and you won't get it.
Also, avoid desserts - Westerners are used to having dessert courses which are often very rich and sweet, whereas the very concept of dessert, or "sweet things" 甜点 as they're known, is quite new here. Given that a most Chinese people don't have a sweet tooth, many are lactose-intolerant, and that the rich, sweet flavours often come from the main dishes themselves, any cakes, biscuits, or puddings you're likely to find here may seem very bland and disappointing.
When you go to a restaurant, wander around the tables first and see what other people are eating and what's popular. A lot of restaurants have picture menus to help. If you don't speak or read Chinese and are not with someone who does then you likely won't find many of the suggestions below (or you might not know it if you have found them) because obviously it's a big country and the dishes don't always look the same - it depends on who cooked it. You could try saving some pictures to your phone and showing them inquisitively to waitresses. Street food is almost always safe and delicious. If you see a tiny, dirty-looking street-side restaurant packed with people, even if it just looks like a hole in the wall, try it.
China also has some major regional styles of cooking:
Northern China has lots of wheat and millet based foods - breads, noodles, pancakes. In the far NorthEast you'll get hearty stews. Across most of Northern China you'll get restaurants and peddlers selling roast lamb and naan bread.
In the Central areas from Hubei and Hunan to Guizhou and Sichuan the climate can be harsh - anything from torrential rains, stifling heat, to freezing rain - and the terrain quite mountainous. These are hard-working, weather-worn people with a tradition for very spicy and strong flavours, and preserved meats, pickles, wines and curds feature quite heavily. The lands are also quite lush and fertile, and so Sichuan food for instance is the equivalent of French or Italian in the West - generally the most famous/popular with lots of special ingredients and techniques.
Further South, Yunnan is the most ethnically diverse province and so boasts its own plethora of food styles. There are influences from sichuan cooking, as well as from South East Asia - ingredients like galangal and lemongrass make an appearance. It's also the land of wild mushrooms.
Heading East from there you come across the food more familiar to Western palettes - Cantonese. Guangdong / Hong Kong has much more to offer than dim sum though. Guangdong people are famous for eating anything. In terms of flavours, it's generally lighter / sweeter and features lots of seafood.
In fact following the coastline all the way back up to Dalian in the NorthEast, passing through Fujian, Zhejiang, Shanghai, Jiangsu, and Shandong, you'll find lots of seafood. Again lots of lighter dishes, compared to Central/Northern cuisines, with more steaming, braising, light soy-based sauces, aromatic vinegars etc.
Ask any Chinese person what they most like to eat and your guaranteed to get the same response - whatever mummy cooks. Assuming you're mother isn't Chinese, here's a list of some of the most popular/well-known dishes you'll find in restaurants, some of which are always winners with the tourists and a few only for the adventurous - you can decide what you want to go for. I've chosen these randomly - just some of my favourites off the top of my head - and listed in no particular order (except the first one... that had to be first!):
#1 GongBao JiDing 宫爆鸡丁
Liked by everyone in the universe - sweet, spicy, chewy, crunchy. It's chunks of boneless chicken fried up with onions and peanuts with dried chilli and sichuan peppercorns. Very easy to find all over the country.
#2 Jiaozi 饺子
Boiled (sometimes steamed) dumplings, usually stuffed with pork and eaten dipped in vinegar and chilli. Very popular in the North, especially in winter.
#3 Hot Pot 火锅
If mummy's cooking isn't their favourite, I bet this probably is. There are various styles but by far the best is Sichuan hot pot - a boiling pot of stock, topped with a few centimetres of boiling chilli oil, floating with dried chillis and peppercorns. You order raw ingredients and cook them yourself then coat in a dipping sauce (usually garlic sesame oil in the South, or thick tahini-like sesame paste in the North). Other styles include Yunnan hotpot, which is more like a chicken soup, and Beijing hotpot, which is just plain water - very bland. Hot pot is available everywhere. MUST try.
#4 Yang Rou Chuanr 羊肉串儿
Roast lamb kebabs, seasoned with salt, chilli, and cumin. The perfect accompaniment to cold beer and found on most street corners throughout the year, especially in the North. In fact the best you're likely to find have got to be on Xian's “muslim street”.
#5 Hui Guo Rou 回锅肉
This translates as "back to the pan" pork. A classic Sichuan dish, it's pork belly that's first boiled, then thinly sliced, then fried until slightly crispy along with onions and fermented bean paste. Salty and delicious, and very commonly found all over the country.
#6 MaPo DouFu 麻婆豆腐
If you haven't eaten tofu in China before then you've never eaten tofu, and this is China's most famous tofu dish. Again it's very commonly found in many restaurants all over the country. Made well it's like a rich bolognese - silky, creamy cubes of tofu to offset a fierce sauce of chilli, bean paste, sichuan pepper, and ground beef.
#7 Shanghai Soup Dumplings 汤包
They most commonly come in a small sizes - a good Shanghai breakfast is a bamboo steamer containing 10 of these babies. Be careful when you first bite in - it's a little pastry sack full of hot soup with a meatball in the middle.
#8 Soy Milk 豆浆
Nothing like the stuff that hippies drink from cartons - the real deal is rich and creamy. It does a good job of taking the heat off a spicy meal... A good way to settle your stomach after a night's drinking too.
#9 DanDan Noodles 担担面
The classic Sichuan noodle dish. Sweet, salty, sour, spicy, all in one. Usually only found in Sichuan restaurants or sold by street peddlers in many parts of Sichuan.
#10 Shui Zhu Yu 水煮鱼
Usually grass carp very gently braised in hot oil that's been infused with chilli and peppercorns. It's not spicy but the flesh is very soft and fragrant.
#11 GuiLin Rice Noodles 桂林米粉
Very popular, although I'm not a fan myself. It's a rich soup with very thin, delicate rice noodles. Comes in many flavours with various condiments. Found everywhere.
#12 Numbing Spicy Crayfish 麻辣小龙虾
A classic dish from Chengdu's outdoor BBQs and beer gardens. They're a bastard to crack into - use your hands and maybe wear gloves - but once you're in then it's like the wizard of happiness has cast a spell on your mouth.
#13 Dai Wei GanBa 傣味干吧
Yunnan province has a great variety of delicious dried meats, most commonly pork, beef, and yak. Yunnan restaurants can be found all over the country though and this is a popular dish - 'Dai' flavour shredded dried beef. The Dai are one of Yunnan's ethnic minorities. Conveniently their name rhymes with Thai and their dishes often contain the same ingredients you might find in Thailand - thai chillis, lemongrass, mint, galangal, and coriander.
#14 Nan Bao Rou 馕包肉
A classic dish from China's primarily muslim North-Western region, Xinjiang. It's Lamb stew served on top of a naan bread. This can be found in many restaurants in Northern China, especially Beijing and Xian. Another notable Xinjiang dish is DaPanJi (大盘鸡, literally "big plate chicken") - basically a big chicken curry with potato chunks.
#15 Stinky Tofu 臭豆腐
Even if you don't try this, the chances are you will inadvertently smell it because it's a favourite street food throughout the country. It's tofu that's been fermented until very pungent then deep or shallow fried and often served with a salty and spicy sauce. It tastes a lot better than it smells.
#16 Steamed chicken feet 凤爪
Chances are, you've probably had Hong Kong-style dim sum before so you probably know what to look for. This is definitely worth a try though - just suck off the skin and the meat and spit out the tiny bones. Delicious!
#17 Cross-the-Bridge Rice Noodles 过桥米线
Yunnan's most famous dish - the story goes that a young lady was worried about her husband's lunch going cold before she had chance to cross the bridge and deliver it to him. She decided to keep the soup separate from the noodles and mix it all up once she'd arrived. Hence when you order this you'll get a bowl of thick, soft rice noodles (a bit like Udon noodles) either with soup or with soup to be ladled on - the various condiments (dried meats, mushrooms, pickles, roots, herbs) will be laid out to be added separately.
#18 Dong Po Rou 东坡肉
This one has a story too - a famous poet got drunk and left his dinner cooking too long. He sobered up when he smelt the aroma and was pleasantly surprised to find that it was actually delicious. Indeed it is delicious - big chunks of pork belly braised very slowly in wine, soy sauce, and sugar until dry. The meat is packed with flavour and collapses between your chopsticks.
#19 Pickled Cabbage Fish 酸菜鱼
China has a huge variety of pickles. Like Korea has Kimchi, there is a type of pickled cabbage that's widely available everywhere in China. It's used to make the base of this soup in which a whole fish (usually grass carp) is poached.
#20 Steam-Pot Chicken 气锅鸡
You're unlikely to come across this unless you're in Kunming but if you get the chance to try the best chicken soup you're ever going to taste in your life then take it. The "Gas Pot" is shaped so that the steam that foolishly tries to escape is condensed back into the soup so no flavour or moisture is lost. Concentrated chicken.
#21 Old Beijing Yoghurt 老北京酸奶
Again, common throughout the deserts and grasslands of Northern China but found in every corner / street side shop in Beijing. It's quite sour and more of a drinking consistency. Yoghurt probably wasn't something that came to mind when you thought of Chinese food was it?
#22 Hairy Crabs 毛蟹
Very popular in Shanghai. You can have them boiled, steamed, braised, stewed, fried... they're only small but the meat is very succulent. They're not actually hairy apart from the bit around the outside of their claws - looks like they're wearing boxing gloves.
#23 Egg and Tomato 西红柿鸡蛋
Just 2 simple ingredients but real comfort food. The salty egg with the sweet and sour tomato goes very well with steamed white rice. Surprisingly tasty. Probably every restaurant in the country can cook this for you. The best part? This will only set you back about US$2.
#24 Pork and Preserved egg Rice Porridge 皮蛋瘦肉粥
Whether you're cold, have an upset stomach, or just want a lip-smacking, rib-sticking start to the day, try this for another classic comfort food. The rice is cooked very slowly in lots of water until it becomes a glutinous soup with that smoky rice flavour. It comes with all sorts of seasonings and flavours (sweet and salty) but my personal favourite is this one which contained ground lean pork and preserved eggs. Rice porridge is sometimes translated on restaurant menus as “gruel” but don't let that put you off.
#25 Guo Bao Rou 锅包肉
If you're missing your familiar sweet and crispy Chinese food then try this - it's a classic NorthEastern dish - thin slices of lean pork covered in a thick and chewy batter then deep fried and served with a sweet glaze with spring onions and coriander. Generally only found however in restaurants that specialise in NorthEastern food, which are quite common in Beijing.
**EDIT: Can't believe I left out this one - definitely a MUST try!**
#26 Dry fried beans 干煸豆角 / 干煸四季豆
Everyone who tries this is wide-eyed on the first mouthful... and it's always the first thing my dad requests when he visits China. The beans are flash fried in super hot oil with ground meat, sichuan peppercorns, and dried chilli. The beans remain quite crunchy with blistered skin. The flavour is salty, spicy, and numbing (from the pepper).
** 2014/12/03 Additions suggested by Thomas Wehner**
# 27 Big Plate Chicken 大盘鸡
Basically, a massive plate of chicken curry, usually containing a randomly chopped up whole chicken - bones, feet, head and all - with potatoes, green peppers, and wide, flat, wheat noodles, all in a rich tomatoey gravy. This dish, like #14 Nan Bao Rou 馕包肉 above derives from China's predominantly muslim region - Xinjiang 新疆 - and so can be found in most muslim restaurants (i.e. the ones with the arabic signage, green decor, minarets etc.) throughout the country, but especially across Northern China and also down in Yunnan province where there is a large muslim population. Excellent served with toasted naan bread.
# 28 LanZhou Pulled Noodles 兰州拉面
I know there are already some noodle dishes listed but this is THE original noodle dish. The city of Lanzhou, the provincial capital of China's poorest province, Gansu, is famous for is long history of hand-pulled noodles. If you get the chance to see a master at work rolling out the dough, stretching and pulling it apart, folding it back and repeating, until his arms are coiled with these delicate wheat noodles, then definitely try a bowl. The Chinese name for pulled noodles - “lamian” - is the origin of the more familiar Japanese word “ramen”, so if you like ramen, you'll probably like these. Served in a thin, light but flavoursome beef broth, the noodles are topped with a few slices of broiled beef, chopped spring onions, and coriander, with the option of chilli added to taste.
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