Arriving to Girona from the airport of Barcelona can take a while. There are very few buses, usually not worth checking, so you'll need to take a train. A taxi may cost you around 200 euros.

First, you need to arrive to Barcelona's main train station, Barcelona Sants, and then take a train from there.  The main difficult is that there isn't (yet) a train from terminal 1 (the usual arrivals) to Barcelona Sants. There is a train which departs from terminal 2, the old terminal, to Barcelona Sants, and apparently the connection to terminal 1 is (still) under construction.

So, first step: how to arrive to Barcelona Sants, the main terminal and departure for trains to Girona. You have the choices:

• a taxi to Sants should cost you 39€ (maybe plus baggage)
• you may take a shuttle bus from terminal 1 to Barcelona (6€). Leave in Plaça d'Espanya, and then walk (20 min)  to Barcelona Sants. If you have some luggage you may try the metro, but probably I would recommend you a taxi, which should cost you less than 10€
• shuttle bus from terminal 1 to terminal 2 (free). Then take the train to Barcelona Sants. Timetable here (aeroport to Sants, cheap), at the time of writing they leave every 30 min, at the hour+08 and 38. You'll need some time to buy the tickets!

Okay, now you are in Sants. Next step is taking a train to Girona. You have two choices:

• normal train, takes about 1h 30 min. Beautiful views, but sometimes delayed (not so often, but you may be unlucky)
• fast train, 35 min. For 17€ you go faster and, more important, more reliable. But... the train may be full. If you buy the ticket ahead, you risk missing it if your fly is delayed.

Check both here, in renfe. From:  BARCELONA-SANTS TO Girona. MD and REGIONAL (very slow) are normal trains. AVANT and AVE actually refer to the same train, only that AVANT is much cheaper.

My advice is (easier, maybe slower): shuttle to terminal 2, then train to Sants. Once you are waiting for the train in terminal 2, you'll know at what time you'll arrive in Sants and may buy a ticket for the fast train. It may be worth, even if it's not the first train to leave. Fast trains are more convenient, only that maybe more standard (if you are looking for the real Catalonia)

Good luck!