Despite generally being a small country, the weather patterns in Sri Lanka are pretty complex as it always has two monsoons. The low season (May-Aug) is called the Yala monsoon and brings rain to the South and West coast. During this time, places in the North and the East have the best weather but generally speaking, the prices are a lot lower during this period. The shoulder season runs from (April-September) and is generally a good time to visit as the temperatures are not too hot. Peak season runs from (Dec-March) and is when the southern coast is the busiest.
So in a nutshell, the best time to visit the west and south coasts and hill country is from December to March, while the best weather on the east coast is from April/May to September.
