The locals say that you haven't been to Koh Samui unless you've visited the Big Buddha Temple on the north of the island, and it is worth paying your respects there as part of a day out, and most tours of the island will include a trip around the temple and gardens.
The busiest beaches are Lamai and Chaweng, which are both on the east coast, and these are well served by shops and have excellent facilities. If you would rather find a quiet spot to relax, then there are many secluded coves and tiny bays around the island that remain empty throughout the days.
With a heavily forested interior, Koh Samui is a great place to go hiking, and there are several lovely waterfalls that attract tourists, and are a great place for a quick dip after a hard slog through the humid jungle. Hin Lad Waterfall is the most famous, and is about 4KM from Nathon Town, the islands capital.
Thai massage is very popular, and outside the hotels, there are many spas, which offer a huge variety of treatments to suit every need.