Miyako is unusually flat for a Japanese island, which makes cycling a genuine pleasure. Rental shops near the airport and in Hirara town rent out bicycles, scooters, and cars; a scooter gives you the most flexibility for reaching the smaller beaches and capes. Bus services exist but are infrequent and don't cover the island well outside of Hirara — don't rely on them for beach-hopping.
A rental car is the most practical option if you have luggage or are travelling with others. The island is small enough to drive around in a few hours, and all the main beaches have car parks. The three bridges to Irabu, Kurima, and Ikema islands are all open to traffic and toll-free.