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.
Takachiho (高千穂) is known as the land of Japanese Mythology (日本神話). There are lots of shrines and places that are related to Mythology. And also Takachiho Gorge is one of the “Must-See” spot in southern Kyushu area.