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.
Isaburo (いざぶろう) and Shinpei (しんぺい) are same local trains. The only difference is the direction. Isaburo is the train that departs from Hitoyoshi (人吉) to Yoshimatsu (吉松). Shinpei is operated from Yoshimatsu to Hitoyoshi. Both Isaburo and Shipei are the name of the key persons when the construction of this section was completed. Isaburo is the named from Isaburo Yamagata (山縣伊三郎) who was the minister of transportations and telecommunications. Shinpei is named from Shinpei Goto (後藤新平) who was the persident of Japanese Government Railways.
These trains are the local trains. But most of seats are reserved seat. And this train is not just local train but the cruising train and is operated by specialized car. This is one of the best local train in Japan. There is a crew attendant on the train. She shows you the scenery of Hisatsu line (肥薩線). Of course you don’t have any restriction to take this train by Japan Rail Pass.
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.