You can get to the resort part of Mykonos by bus. The bus stop is located right at the exit of the terminal building, but the buses from the airport are intermittent.
If you plan to rent a car right at the airport, you should take the road Επαρ.Οδ. Μυκόνου (the distance is about 4 km). Mykonos Airport has terminals of Avis (tel: +30-22-890-24-171), Hertz (tel: +30-22-890-22-890-22-890), Sixtie, Meridien Tours (tel: +30-22-890-23-173).
But it is best to use cab services - the parking lot is located at the exit of the terminal. The price of a cab fare from the airport to the center is 6 EUR.
