Probably the best way of getting around if you have the appropriate driving license. There are car rental companies in Miyanoura, Anbo and near the airport. You can rent one from about 5,000 yen for 24 hours.
Public buses run about once every hour. Clockwise from Nagata to Kurio, and a few to Ohko Falls and back. There are buses to Shiratani Unsuikyo and Arakawa Traihead too. It’s worth getting a Yakushima Kotsu bus pass (one-day/three-day/four-day ¥2000/¥3000/¥ 4000) if it is your main mode of transport.
Rental bicycles are available from 300 yen/hour or 1,500 yen/day. Motorbikes are also available for rent from 2,000 yen/hour or 3,000 yen/day. You Shop Nangoku has shops both in Miyanoura and Anbo.