How long time-wise and how expensive money-wise are the following options for getting between central Bilbao and central San Sebastian?
Hi Alex. Like Valerie said, there's a direct bus from the Bilbao airport to San Sebastian. But if you'll be in central Bilbao, there are more frequent (and cheaper) buses. They leave from Termibus and arrive in San Sebastian at Pio XII Plaza. From there, it's a quick taxi ride or 15-20 minute walk to Parte Vieja where all the pintxos bars are.
There might be a train between Bilbao and San Sebastian, but for your time and money, the bus is the way to go, for sure.
The most common way to get to San Sebastian is the Pesa bus (*). It is direct and leaves from the Termibus station, easily accessible by metro (San Mamés stop), bus, tram or walking. There are buses almost every hour, the ticket costs 11,75€ and it takes 1h 15min. Once you arrive at the Bus station (well, not really a station, just the street) from the Pio XII square, which is in front of the bus stop, you can take a city bus to almost anywhere.
Train has a scenic view, but it takes too much longer.
Taxi would be too expensive.
(*) Keep in mind that San Sebastián is Donostia in Basque, so the bus will surely indicate so.
(**) If you think you will travel a lot by public transport consider purchasing the Barik card.