A lot depends on your interests!!
There is a lot of time though, so visiting Yakushima would be well worth it. It rains a lot, but there's nowhere in Japan like it.
Nagasaki is really the most interesting city on the island in terms of history and varied points of interest. 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.
http://www.ngs-kenkanren.com/eng/http://www.at-nagasaki.jp/foreign/english/http://www.youtube.com/watch?v=QdWRyHGW3rkBe 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_JnkVInTUFor Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. Here is one site that's a good guide:
http://www.thejapanfaq.com/fukpic.htmlThere are also many useful videos to give you a preview:
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...Be sure not to miss the Nanzoin Temple with its Great Buddha - it's a hidden jewel that few even know of:
https://www.youtube.com/watch?v=lT4VQFmHFyMKitakyushu has a few sights of interest also. It's Kokura Castle is one of the better ones, the view from its Moji wharf and preserved buildings are quite historical, and the view from atop Sarakurayama is one of Japan's best and rarely mentioned:
https://www.youtube.com/watch?v=Hl1pCW9IbbkOita has some nice sites too. Beppu is famous for its hot springs, Hells Tour, plus monkey park & aquarium. See:
https://www.youtube.com/watch?v=K2NNi8MLcXMhttps://www.youtube.com/watch?v=zepPfkuqhokhttps://www.youtube.com/watch?v=cBnZneDsKqQKumamoto is famous for its castle, Suizenji Garden (a must-see), Mt. Aso and Kurokawa hot springs.
http://www.visitkumamoto.com/https://www.youtube.com/watch?v=Tyh_nKrv5tYhttps://www.youtube.com/watch?v=K9xyXmU8U1gNote that you can't get to Kurokawa by rail.
Kagoshima is great to see the live Sakurajima volcano, hiking, and hot sand baths such as on Ibusuki. Plus the Senganen garden has a superb backdrop view of the bay and the volcano. See:
http://www.kagoshima-kankou.com/for/https://www.youtube.com/watch?v=LxDmLuq6ssIhttps://www.youtube.com/watch?v=lpnDK_Wux9QPlus Miyazaki has some very pleasant beaches, and its Takachiho Gorge is remarkable (but you can't get there by rail).
http://www.miyazaki-city.tourism.or.jp/en/http://www.kanko-miyazaki.jp/english/https://www.youtube.com/watch?v=H49x41ViFJkFor comprehensive info on Kyushu, see:
http://www.visitkyushu.org