From the airport, you may travel by bus to Gdańsk Główny (the city center) or to Wrzeszcz (one of the main city districts). From both places you may take a suburban train (SKM) or a tram to your destination. Alternatively, you may take a taxi. We advise you not to accept offers from taxi drivers who come inside the terminal building. Instead, please go to the taxi rank just outside, where authorised taxis belonging to the firm City Plus Neptun operate.
http://www.airport.gdansk.pl/passenger/transfer (information about transfer to the city)