The Miyako Airport is reachable with direct flight from Tōkyō (2,5-3 hours), Ōsaka (2 hours) and Naha (45 minutes) airports.
You can get to Miyako city by stepping on the Miyako-Kyōei bus but, since the service is quite limited, taking a taxi could be better in certain cases (around 1500-1700 yen).
To move around we suggest to rent a car: you can do it directly at the airport and start driving through the island from there.
There is no need to worry about traffic, even during the holiday peaks (however in this case we suggest to reserve in advance).
You can also use a bicycle: there are some shops where you can rent one in Miyako and start exploring the island without haste, especially recommended if you can spend at least 2 or 3 days on the island.
Miyako-jima is 25 km long and 15 km wide, riding a bike through it is nice experience!