Sri Lanka has two separate monsoon seasons hitting two separate coasts, which is the one fact that makes the whole "best time to visit" question answerable.
The Yala monsoon runs roughly May to September and soaks the southwest and hill country Colombo, Galle, Ella, Kandy. The Maha monsoon runs October to January and hits the northeast — Trincomalee, Jaffna, Arugam Bay.
Practically: December to March is the safest window for a first trip, since the west, south, and hill country are all dry. If you are chasing surf on the east coast specifically, April to September is when Arugam Bay comes alive instead.