Most people will take a the Airport Shuttle
to Colombo fort, then either the train or bus to Kandy.
There is a less well known way that will save time and money. grab a taxi from (if a lot of luggage) to Gampaha station then take a train to Kandy
Train from Gampaha first train at 06:30 last train 18:18 approx 9 trains during the day.
Via Bus to either Negombo or Colombo Fort:
Negombo take the free free shuttle bus
to Katunayake bus stand (runs about 3 times an hour) from the airport.There is not much room for luggage - max a big backpack - on the local buses. From there you can get the bus to Negombo.
Alternately take a Taxi to the Katunayake bus stand.
Then catch a bus from Negombo bus stand to Kandy.
No night service and no AC Express buses the every 10-15 minutes by day.