I visited the area in June 2016; things could be different earlier in the year.
San Sebastian is pretty and the beach is magnificent (but it won't be beach weather), but the small historic area is packed with tapas bar and tourists. I found it not a terribly pleasant environment as a relatively introverted solo traveler. It was hard to find a place to eat lunch sitting down because of the mobs. Definitely see it (there's a good museum), but I wouldn't want to spend much time there.
Very good side-trips from SS make it a good base even though I didn't need a full day to see everything I wanted to in the city itself (and I walked around a lot):
Hondarribia: Largish coastal town with both a fishermen's quarter and an older historic area on a hill. Worth a good chunk of a day.
Zumaia/Getaria/Zarautz, a trio of former fishing villages now dedicated to tourism but not overrun. Can all be seen in one day from SS via public transportation.
Bilbao: To me much more interesting than SS. Several museums and a larger historic area than SS's. Also some very nice late-19th/early-20th century buildings. Few tourists except around the Guggenheim. I found Bilbao an extremely pleasant city just for walking around. Bermeo is a good side trip; it's a former fishing village--probably still a fishing village with an overlay of tourism, though I seemed to be the only tourist there on the rainy day I chose to visit.
Vitoria-Gasteiz: capital of the inland part of the Basque Country. Large, hilly, atmospheric medieval district; the climb is escalator-assisted. Several museums. Cathedral was undergoing major reconstruction during my visit, but there was a good English tour offered. Excellent side-trip to the hill town of Laguardia is possible even by public bus, though easier by car. You're now near the La Rioja wine country if you have any interest in wine (I do not).
Lekeitio, another former fishing village now discovered by tourists, is located between SS and Bilbao. I haven't been there, but it is mentioned in all the guidebooks.
The local-train trip between SS and Bilbao goes through some lovely countryside but is extremely slow. I recommend it for its sightseeing value, but you'd want to return by the much faster bus.
