This would depend on for how long you're planning on staying there and what you're into.
If you won't have time to get out of the southern coast, must do places around there are, Galle (Dutch Fort, Architecture, Stunning Boutique hotels), Hikkaduwa (Party Town - careful here - but tourists are usually fine), Unawatuna (Awesome beach), Mirissa (Whale watching), further down you get Tangalle (still somewhat untouched compared to other parts of the southern coast. Comes with caution - again tourist incidents are rare).
Certainly spend a couple of days in Colombo. You can see how this colonial city is transforming in to a modern city..
If you have more than a week, it's highly recommended you go to the High country, Kandy and Nuwara Eliya. Lovely mountains, lakes (somewhat like NZ in the summer) with preserved British architecture.
If there's more time, make a visit to the ancient capital cities Anuradhapura, Sigiriya, Dambulla. And of course the eastern coast; Trincomalee, Pasikudha, Arugam Bay (one of the best surfing spots).
Happy to help if you need more details. PM me.
Cheers..