I hit Kumamoto, Nagasaki, Beppu, Kagoshima, and Fukuoka in five days 12 years back- so keep in mind that my info is old. My interests intersect yours with History, and Experiences/sights. I really enjoyed reading up on Saigo Takamori as most of his story happens in Kyushu and he is a historical hero in Japan. Here were the highlights for me:

Kumamoto: Restored castle, park replicating the Tokkaido road, Musashi Museum (all in Japanese)

Nagasaki: Would spend 2 days minimum- Best A-Bomb museum & cool peace park, Dejima re-creation (major historic significance), great views up the ropeway, great city to walk around. Really regretted keeping my visit to one day.

Beppu: Did the touristy seven hells tour- not necessary but fun. Did the old hot sand onsen as well.

Kagoshima: Sakurajima (not good to do on foot back then), circular bus with lots of great historical sights re: Saigo Takamori.

Fukuoka: Hakata Folk Museum was great, visited the temple where supposedly green tea was first brought from China (wasn't that impressive beyond historical context). Probably could have done something better here.