Hi,
What to see depends a lot on your interests of course, but whatever it is, Kyushu has a lot to satisfy nearly any traveler.
For your Fukuoka itinerary, it is easily doable; in fact it looks rather slow unless you like to linger and browse. There is a good one day itinerary you might want to look at on:
http://www.thejapanfaq.com/fukpic.htmlwhich mentions a lot of good sights. Be sure to try the tonkotsu ramen at least once - there are many yatai (stalls) in Nakasu and Nagahama. The former is more scenic but the latter has more of where the locals eat.
In terms of history and varied points of interest, Nagasaki is really the most interesting city on the island. It is really worth a couple days to see the city. 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_JnkVInTUHuis Ten Bosch will either really impress you or leave you bored. It really does capture the feeling of being in Holland, and you will likely be there in tulip season which is the best. But some find it just a nice Potemkin Village.
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_Wux9QFor Beppu hot spings, you might look at:
http://www.japan-guide.com/e/e4701.htmlThere is more than the hot springs there of course; the Hells Tour is a unique sight and the monkey park/aquarium are good also:
https://www.youtube.com/playlist?list=PL7LcPcmoSg-hOggZamc2lUNTr46ApKX...Yufuin will be quite different than Beppu. Some find it relaxing, others find it dull. There is some hiking there as well, but you may find it is not worth a full day.
For your day 8 you could easily squeeze in some more - you might consider going to Yanagawa and riding the canal boats there.
For your last day, you might consider Kitakyushu. Kokura Castle is one of the better ones, plus the Moji Retro area has some interest. You could even venture into Shimonoseki and see some good sights there. Plus there is Sarakurayama, a nearly unknown gem in Kitakyushu with one of the best mountaintop nighttime views in Japan:
https://www.youtube.com/watch?v=Hl1pCW9Ibbkhttps://www.youtube.com/watch?v=6y4K6erVyvAHope you enjoy your trip.