I recently went on a 6-day, 5-night trip around Northern Kyushu, where I visited Fukuoka, Kumamoto, and Nagasaki. Truth is, just in Nagasaki alone, 6 days may not be enough because the prefecture is huge and there are quite a number of historical places to visit - from war memorabilia to Catholic churches.
Traveling within Japan can be so much fun, and can also be quite expensive. More so when traveling around the Kyushu region,where there are so many places to see. However, there are ways to enjoy Kyushu, specifically Northern Kyushu, without actually breaking the bank.
Book your flights early. Surely you have already heard this tip from somebody else. This is nothing new. I highly recommend availing the JAL Sakitoku fares. You may reserve your flights as early as 6 months prior without having to pay yet, so you may cancel the reserved flight without any hassle. You'll only be required to pay 2 months prior to departure.
Same as with flights, book your hotels early. There are reputable travel agencies all over Japan that can provide you with great travel packages at reasonable costs. However, unless you can speak and understand Japanese, you probably won't be able to get a good bargain. I love using Agoda.com for booking hotels for two reasons: (1) their rates are quite cheap and they come with perks such as free cancellation and pay later options; and (2) they have the most number of tie-ups with frequent flier or mileage programs of major airlines like Delta Airlines and ANA, making it possible to earn miles on even the cheapest business hotels.
Get the SunQ Northern Kyushu Pass. For only 8,000 yen, you have a 3-day ride-all-you-can privilege on all buses (including the highway buses) around Northern Kyushu. This is a cost saver since a one-way bus ride from one prefecture to another costs around 2,000 yen minimum. What's more, you'll get to enjoy wonderful views of Kyushu's mountains and rice fields, which you probably won't get if you take the trains or the Shinkansen. More details can be found here: http://www.sunqpass.jp/english/pass/index.html.
When in Fukuoka City and you would like to visit another prefecture, like Kumamoto or Nagasaki, make sure you leave early, to avoid traffic and to have more time going around. Traffic builds up along the highway (before Tosu exit) around 9 or 10 am, so a bus trip that would normally take two hours could last 3 hours if you're not so lucky.
When in Nagasaki City, get the one-day street car pass, so you can hop on and off the street cars as often as you want within a day for only 500 yen. This pass cannot be bought in the street cars, you have to buy them at hotels or at the tourism information center at Nagasaki station. Most hotels can only accept cash payments; if you prefer to pay using credit or debit card, I recommend getting the pass at JR Kyushu Hotel Nagasaki.
If a trip to Gunkanjima is in your itinerary, it is best to reserve slots early, as these ferry trips are quite popular and slots gets filled up fast.
One more tip: you may scrimp on the flights and hotels but it's best not to scrimp on the food. Just as plentiful as the sights in Kyushu are the sumptuous food on offer. Make sure to have seconds (or kae-dama) of your ramen noodles in Hakata, and try the shippoku in Nagasaki.
Enjoy!