Japan Railways Shinkansen operates a train from Fukuoka to Kagoshima hourly. Tickets cost $55 - $80 and the journey takes 1 h 35 min. Alternatively, Japan Air Commuter flies from Fukuoka to Kagoshima 3 times a day.
How to get from Fukuoka to Kagoshima
From Hakata train to Kagoshima-Chuo. 1 h 35 min ¥7537. From Fukuoka drive to Kagoshima. 3 h 13 min ¥4013. From Hakata subway to Fukuoka airport, fly to Kagoshima airport, taxi to Hayato(Kagoshima), then train to Kagoshima. 3 h 11 min ¥28485.