Take a bike around the island. Shark Bay is a good place to go snorkelling for the day (don't worry, they are only small black-tip reef sharks, harmless to humans). Ko 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 travel them. While there are legally no private beaches in Thailand, some are relatively inaccessible by land making a paid longtail boat a necessity, and some are "owned" or only accessible via "private roads" that charge 100-200 baht entrance fees..
However, to see the best places, go by foot or longtail as some places 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.