The Kagoshima Airport is fairly small with only several domestic destinations and a few international ones as most travelers use the larger Fukuoka Airport. Several airlines connect Kagoshima to Tokyo - Haneda Airport and Japan Air Commuter offer several flights to other domestic destinations.
Kagoshima Airport is not a large airport so there are not many flights departing here. There are only a few destinations that you can reach from Kagoshima Airport, most of these are operated by Japan Airlines. Many people take a flight to Tokyo and transfer to another flight there.
The Kagoshima Airport is located about 30 km northeast of the large Kagoshima-Chuo bullet train station.
Taxis are available but since the airport is quite far away from the city a better option is to use the airport bus (1200 JPY one-way) which will take you to the central train station. From there you can switch to public transport or get into a taxi to get to your final destination.
All prices quoted here were found in July 2017