I wouldn't bother taking the train. (It involves getting a bus into Girona first). It's much simpler just to take the express bus.
Timetable from airport:
http://www.sagales.com/pubdocs/602_aeroport-gi rona-costa-brava---barcelona-(estacio-nord)-a- partir-del-2008-06-12.pdf
Timetable from Barcelona:
http://www.sagales.com/pubdocs/603_barcelona-( estacio-nord)---aeroport-girona-costa-brava-a- partir-de-2008-06-12.pdf
Once at the bus station, you can get a taxi to your accommodation. Alternatively, there's an excellent, cheap integrated public transport system (metro, buses, train and trams) that can get you wherever you want to go:
http://www.tmb.net/en_US/home.jsp
Chris