Take a bike around the island. Shark Bay is a good place to go snorkelling for the day (don't worry too much about sharks, they are only small black-tip reef sharks). 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.