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.
Most flights from Kagoshima Airport are to Tokyo and to Osaka but there are many more flights to other destinations as well, most of them by Japan Airlines. Every day there are flights to 5 destinations from Kagoshima Airport. Here are the most popular destinations:
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