Kikai Island is a small island located in the Amami Islands, just to the east of Amami Oshima. While technically a part of Kagoshima, due to a long and varied history of being conquered, culturally Kikai is a blend of Okinawan, Amami, and Kagoshima traditions.
Kikai has a small airport located in the main town on the island which has daily flights to and from Kagoshima airport, and Amami Oshima airport.
All routes in and out of the island are owned by JAC, an affiliate of JAL.
A one-way ticket from Kagoshima will cost you approximately ￥24300. A round-trip ticket is approximately ￥44200 total, or about ￥22100 each way, making it slightly cheaper, but still pretty expensive.
Bouncing out of Amami is possible, though no cheaper.
The ferry is a much better bet unless you have way more money than time.
Make no mistakes, Kikai is small. The costal road surrounding the island can be walked in 6 to 8 hours depending on your pace, meaning that getting to any specific point on the island will probably take no more than 2 or 3 hours on foot. This of course assumes that you have a specific point that you are trying to get to. Most of Kikai outside of the main town is endless sugar cane fields.
If you need to get somewhere specific and don't feel like walking for hours (especially in the summer heat), there is a bus service (air-conditioned) operated by A-Line Ferry.
The fare is ￥300 to anywhere on the island.
There are 3 bus routes on the island. Two of them, 北本線 (north line) and 南本線 (south line) follow the coast road clockwise and counter-clockwise respectively. The third 中央線 (center line) runs into the island interior, and is somewhat less useful unless you're looking for a way to get up into the mountain(s).
The bus is populated mostly by 70 to 90 year-old women in varying states of decay either going to, or returning from, the hospital. If you're looking for a place to hear some of the island's endangered language, the bus is a great place to do it.