What to see depends a lot on your interests of course, but Kyushu has something for nearly any traveler. There are a lot of places of course - but unless you are sure you will not be back in Kyushu for a very, very long time, if ever, it would probably be better to concentrate on the northern half of the island and use a 5 day JR Northern Kyushu Rail Pass.
Regarding onsen, I think most people would agree that Kurokawa is the best on the island. Unfortunately it is isolated and you can't get there by train. But the outdoor baths in the serene forested atmosphere is really special. Some other possibilities are Beppu, Yufuin, and Unzen.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide:
http://www.thejapanfaq.com/fukpic.htmlDon't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely
free!
https://www.youtube.com/watch?v=lT4VQFmHFyMThere are also many useful videos to give you a preview:
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...Look into a Fukuoka Tourist City Pass also:
http://yokanavi.com/access_tourist_egThe Rakusuien and Yusentei Gardens are also very beautiful.
https://www.youtube.com/watch?v=yLUJeYi57kshttps://www.youtube.com/watch?v=AbsU4N99Qu4Nagasaki is really the most interesting city on the island. It is worth a couple days to see the city, and another day to see areas nearby like Shimabara/Unzen or whatever interests you. Gunkanjima, while it takes 3 hours, will blow your mind. Be sure to see Mt. Inasa at night if the weather is clear, and try the Nagasaki chanpon and saraudon! Also the Koshibyo temple is an often overlooked shrine that is quite unique in Japan:
https://www.youtube.com/watch?v=Mu_JnkVInTUKumamoto is famous for its castle, Suizenji Garden (a must-see), Mt. Aso (crater area currently closed).
https://www.youtube.com/watch?v=K9xyXmU8U1ghttps://www.youtube.com/watch?v=Tyh_nKrv5tYOita has some nice sites too. Beppu is famous for its hot springs, Hells Tour, plus monkey park & aquarium.
https://www.youtube.com/watch?v=K2NNi8MLcXMhttps://www.youtube.com/watch?v=zepPfkuqhokFor comprehensive info on Kyushu, see:
http://www.visitkyushu.org