Three good day trips run from Kagoshima:
Ibusuki, an hour south by the Ibusuki-no-Tamatebako sightseeing train, is famous for its sand baths — you lie buried up to the neck in steaming volcanic sand on the beach, sweating into a towel.
Chiran, an hour and a half by bus, is the best-preserved samurai street in Kyushu, with seven small Edo-period stroll gardens still owned by the original families. The town also has a memorial museum to the kamikaze pilots who flew their last sorties from a wartime airfield on its outskirts; the museum is sobering and worth the trip.
Kirishima, the mountain range north of the city, has the volcanic crater lake of Onami-no-ike, the old Kirishima-jingu shrine and a string of onsen towns. The toll road over the ridge gives the best mountain views in southern Kyushu.
For the longer trip, the cedar forests of Yakushima, three hours by jetfoil from Kagoshima port, deserve at least two nights.