From Helsinki-Vantaa airport there are direct long-distance busses to Turku, but if you prefer to take the train you will have to go to Helsinki city center first.
Local busses from Turku airport to the city center (bus 1) run every 20 minutes. A single ticket costs 2,5€ and a travel ticket valid for 24 hours 6,50€. The timetables for summer will be posted here as soon as they are published by the city. A taxi from the airport to the city center (about 9 km) costs circa 15€.