Kagoshima

Sakurajima
Kagoshima's temperamental and active volcano can be reached by ferry in a matter of minutes. Face your fear and get up close and personal - just don't wear too much white, as the ash will leave you looking slightly grey. Yunohira Lookout is a two hours hike or 15 minutes drive from Sakurajima harbor and gives you a good view both to Kagoshima harbor and Sakurajima volcano. There are also a couple of other lookout points, where you can take a closer look at the volcano. Because Sakurajima is essentially an active volcano, remember that tourists are forbidden to go within 2 km of the crater.
Tsurumaru Castle
Once the castle of the Shimazu clan who governed the Satsuma domain till the Meiji Restoration, it was destroyed during the Satsuma Rebellion. The only parts remaining today are the wall and the moat. Since it's around so many sights and monument around Kagoshima, it's a nice sight to see and it contains the Reimeikan.
Kagoshima Prefectural Museum of Culture
Reimeikan

Beautiful museum with nice expositions retracing the origin and evolution of the first habitants of the southern Kyushu area till the Meiji restoration. Generic information is available in english but for more details about the differents pieces contained in the exposition, a Japanese reading friend should be brought to fully enjoy all the information available.

Sengan'en Garden
Kagoshima City Aquarium
Io World
The Nanshu Cemetery