no trains from the airport, but you can get bus to gerona and train from there.
have a look here
for more information about transport.
to be honest i would say your best bet would be either a taxi from the rank at about €50 or a transfer company like girotransfer/shuttledirect/alpharooms.
if you get a train from girona centre you then need to change at macanet / massanes, really easy and cheap
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