Bilbao airport to San Sebastian?

I'm shortly travelling to San Sebastian and am flying in to Bilbao.
From what i can gather the best way would be to get the bus from the
airport to the centre of Bilbao and then either get the bus or train
from there to San Sebastian. If anyone out there has any advice as to
what the best way to do this then please let me know. I assume there
isn't a bus directly from the airport at Bilbao to San Sebastian? Also
i have read that some trains can take a long time and stop at every
place on the way. I'd like to avoid this if possible, unless the route
is through spectacular scenery or offers something that should not be
missed.

At present i'm thinking of getting the bus service to Plaze Moyua and
then from there getting a train from the Estacion Atxuri with Eusko
Trenibideak to San Sebastian. Is this the best way? Any comments or
anyone who has made this trip (or lives in the area) please let me
know. I can't wait to visit the region. My girlfriend is working there
and it seems like a great place to see. Also if anyone has any comments
on what to see and do once there please let me know (as any help or
advice would be much appreciated).

Matt