Colombo is the booming Capital of Sri Lanka with some really nice buildings, temples and a great beach area populated by lots of people evenings and weekends.
Kandi is a nice and central city where you can relax and enjoy Sri Lanka. The main attractions are the lake, the Tooth Relic temple and the parks in the direct surrounding area.
Sigiriya is famous for the gigantic rock with ancient carving, elephant foot entrance to a merely accessible path and ultimately the remains of a temple on top of the rock. Every visit to Sri Lanka should include this sacred place. But it is hot out there, so bring your sun hat and crème!
Nuwara Eliya is located in the heart of Sri Lanka, the Tea Country. There are MANY leafs, many leaf pickers and it is very green. Of course you should have a tea and visit a tea factory to learn how the leafs end up in your cup (especially for the British...). Note that there is an interesting train ride from Ella - Nuwara Eliya - Kandi to Colombo.
Dambulla Cave Temple is the largest and best preserved cave temple complex in Sri Lanka. Major attractions include 5 caves that contain statues and paintings and are accessible to the public.
Anuradhapura is a major city and one of the ancient capitals of Sri Lanka, famous for its well-preserved ruins of an ancient Sri Lankan civilization. The temple complex is best viewed by riksja.
Polonnaruwa was once the capital of the Kingdom Polonnaruwa. Now it is an ancient place with a lot of ruins.