Take a bike around the island. Shark Bay is a good place to go snorkeling for the day don't worry too much about sharks, they are only small black-tip reef sharks. Koh Nang Yuan, Freedom beach, Laem Thian, Mango Bay, Hin Wong Bay and Tanote Bay are good day excursions, accessible by road or boat. If you rent a scooter, be prepared to turn around or continue by foot since some roads e.g. to Mango Bay may be in such condition that you need a dirt bike or a car to ride them. Some beaches are private, but a boat can take you snorkeling to those bays also.
However, to see the best places, go by foot as some are inaccessible to motorists. There are a number of good viewpoints around the island such as John Suwan mountain viewpoint in the South of the island, Two View in the centre and Fraggle Rock in the North.