When you go to Kagoshima, Kyushu Shinkansen is the best way to get there. It is fast and the trains come very frequently. But if you have a time, please consider the route from Kumamoto to Kagoshima-Chuo via Hitoyoshi and Yoshimatsu on Hisatsu line (肥薩線). It is very scenic route. JR Kyushu operate very attractive trains on this route.
Hayato no Kaze is the limited express train that runs between Kagoshima-Chuo with Yoshimatsu. You can take this train by Japan Rail Pass with no extra charge. This train is the limited express train but this train is slow on Hisatsu line. Hisatsu line is very scenic route. This train is operated by specialized KIHA 147 series. The fleet of this train was fully renovated and it has very cozy Japanese atmosphere. The attendants are on board. They guide you about the scenery of Hisatsu line. This train stop at Kirishima-Onsen station. So you can take this train when you go to Kirishima mountains.
I will show you many images of this train and explain the details.