From Helsinki-Vantaa airport there is a bus connection to Turku (Turku linja-autoasema). The long-distance buses (called in Finnish pikavuoro) depart either from platform 13 (laituri 13) or platform 14 (laituri 14). Buses run every hour and the trip takes approximately 2 h 30 mins. You will have to change the buses in Espoo IKEA (approximately in 30-40 minutes after the departure). The driver will inform you about the stop, as well as he will move your luggage to the other bus. Tickets are sold by the driver, no reservation is required. The price of a single ticket is about 31 Eur. Here is the timetable of the bus service.
There is also a train connection to Turku. The train is more comfortable than the bus and the trip takes 2 hours. However, to take the train, one has to get to the station first, and the whole trip might get a little longer than by the bus. There are two options: