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 (はやとの風) is a limited express train that is operated between Kagoshima-Chuo (鹿児島中央) with Yoshimatsu (吉松). You can take this train by Japan Rail Pass with no extra charge. This train is not an intercity express train. This train is slow and is operated for sightseeing along Hisatsu line (肥薩線). This train is operated by specialized KIHA 147 series. This train has the true Japanese atmosphere. The crew attendants are on board. They guides 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.
Hayato no Kaze will runs as the seasonal train after March 17, 2018. Please do not expect that the train runs every day. Please access Seasonal trains schedule to find the train operation dates.