Other cities to visit after Barcelona

This might give you some ideas of other cities where to go based on your own interests of the area around Barcelona
http://www.catalunya.com/

There are direct flights from Barcelona to San Sebastian. That is the only mode of transport I would use to get there given the amount of time you have. This official tourism site of the Basque area should give you some ideas based on your own interests:
https://tourism.euskadi.eus/en/

From Barcelona, you can take the AVE High Speed Train (1:30 hr) to Zaragoza in the neighboring Aragon region. Zaragoza has more than 2,000 years of history including Roman ruins of the city walls and amphitheater, as well as the Aljafería, an 11th-century Moorish palace. It also has the famous pilgrimage site of Nuestra Señora del Pilar Basilica, housing the miraculous image of the Virgin Mary. The Pilar Basilica is my vote for the most beautiful exterior of any church in Spain. I would say that Zaragoza is the least touristy of the 5 major cities of Spain.

From Zaragoza you can rent a car and drive to San Sebastian (2:40 hr) and stay there for a few nights too. In between Zaragoza and San Sebastian, I would also stop at the medieval town and Royal Palace of Olite, the seat of the Kings of Navarre. For me, one of the most impressive medieval palaces in Europe. San Sebastian itself is an International culinary capital, and feels very Belle Époque... maybe a bit too posh for my taste. Of course the main attraction are those famous Pintxos bars that line every street.

From San Sebastian you can fly direct back to Barcelona.

You could spend 2-3 days in Girona and day trip to the Dali Theatre in Figueres. I really liked Valencia. That's pretty much my experience in that part of Spain.