There's no train from the airport (though a high-speed rail is planned), so taxi or bus are your main transport options. Taxis are relatively inexpensive and reliable.
Taxi
Go to the counter, tell the staff your destination, including the zone (1-15), and they'll print a voucher with the price. The journey takes between 30-60 minutes, depending on traffic.
Cost: LKR 2,300-LKR 2,700 without air-con, LKR 2,550-LKR 3,000 with air-con
Location: Taxi counter, immediately after leaving customs, on the right
Bus
A cheap, intercity bus service (187-E3) leaves for Colombo approximately every 30 minutes. The journey takes around 30-60 minutes via the Colombo-Katunayake expressway. The buses run regularly from 06:00-18:00 every day. Two buses run from 18:00-19:00, and after that buses will be available based on the number of passengers.
When heading back to the airport: Be aware that only these big blue buses go into the airport itself. Other buses may say 'airport' on them, but stop at a nearby bus station.
Cost: LKR 130
Location: Come out of the airport, turn left, walk to the end of building, cross the road and a big blue bus in the car park. There aren't any signs, so you may have to ask airport personnel.
Tuk-tuk
If you want the authentic Sri Lankan experience, and don't have much luggage, take one of the local three-wheelers. However, the price will not be much different to a taxi.