Kagoshima port train station

what is the name of the train station of the Kagoshima port because we need to take the ferry to Yakushima from there and it's not really clear how to get to the port in Kagoshima

There is no station close to the port. It is easiest to take a taxi from Kagoshima-Chuo Station.

This is the location of the jetfoil port:
http://www.tykousoku.jp/dock/kagoshima.html

This is the location of the daytime ferry terminal:
http://ferryyakusima2.com/terminal.html

This is the location of the overnight ferry via Tanegashima:
http://www.yakushimaferry.com/

I went to Yakushima Island earlier this year on a jetfoil. It was pretty easy to get to the port from the station.

I asked at the information desk at shin-kagoshima station, and they directed me to a specific bus, the number of which I can't remember. The cost to ride the bus was a flat 150 yen. The bus went directly to the port building.

I hope this helps.