There is a great hike you can take around the north end of the island, but that takes most of one day. Give yourself time to really experience this very special place. Rent a car, motorcycle, bike, take a taxi, walk, join a tour, or rent a horse. There are some 100 taxis, and they can take you around but you might enjoy going at your own speed, taking time to see things that interest you. If you rent a bicycle, bring your own bike lock.
Horses look glamorous but they are mostly pretty miserable beasts.
It is more efficient to have a vehicle, in my opinion. You can rent a vehicle through your hotel, or from one of the many places in the village that advertise car rentals. There is no water safe to drink elsewhere on the island.