I spent one day in Kagoshima, and saw many things walking around.
Around the city are statues of historical figures. Most give English, as well as Japanese explanations.
Terukuni Shrine is dedicated to Shimadsu Nariakira, the 28th lord of the Shimadzu family.
Kagoshima Prefectural Museum is a free museum geared toward children, but still interesting for adults. They gave me a gift of volcanic ash of Sakurazima in a little plastic bag.
Kagoshima City Museum of Art did not allow photos inside. The entrance fee is a reasonable 300 yen. This museum was constructed on the ruins of Tsurumaru Castle, the fortress of the lords of Satsuma. The original museum was founded in 1954, and the new building opened in 1985.