This post is aimed at helping people who wish to travel to the Costa Brava by public transport from the airports of Barcelona and Girona.
The new, high speed train line between Barcelona, Girona and Figueres is a fast and easy way to travel between these cities. http://www.renfe.com/EN/viajeros/horarios.html However if your final destination is the Costa Brava coast then either bus or a combination of train and bus are your only options.
As well as the main fast speed train line linking Barcelona, Girona and Figueres there is a secondary slower train from Barcelona which passes through Girona and Figueres to the frontier town of Portbou. http://rodalies.gencat.cat/web/.content/pdf/horaris/R11.pdf
The bus company Sarfa offer routes directly from Barcelona airport to most of the main coastal towns on the Costa Brava http://www.grupsarbus.com/ca/trayectos.html and also from Girona airport http://www.grupsarbus.com/ca/trayectos-girona.html
Alternatively there is the option of taking the train to Girona and then a bus on to your final destination. As well as Sarfa other local bus companies include: Sagales http://www.sagales.com/ and Teisa http://www.teisa-bus.com/
Teisa also have a direct route between Barcelona and Olot stopping at Banyoles and Besalu. http://www.teisa-bus.com/ca/bus-interurba