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.
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!
Polonnaruwa was once the capital of the Kingdom Polonnaruwa. Now it is an ancient place with a lot of ruins.
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.
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.
Ella is a small town in the center of Sri Lanka. It is the starting point for two attractions in these hills: little Adam's peak and the scenic train to Kandy. The five arches bridge, waterfalls and rice fields are nearby.
Old and interesting beach city with a famous (Dutch) fortress with walls and everything. One of Sri Lankas ´must-do´ locations. Note the nice beaches and fishermen on stilts a little more south...
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.