Summary
Known as the Satsuma domain in antiquity, Kagoshima's will excite, especially if you're into smoking mountains, wrecked-up castles, and alchohol.
Sakurajima is a volcanic mountain that perpetually gives off smoke, erupting now and then. Kagoshima is famous for shōchū, one of the most traditional alcoholic beverages in Japan. If you're there at the right time of year, you can see the Umbrella Burning Festival, the Sand Festival, or Spider Fighting.
Kagoshima may be hard to get to, but it's worth the effort.