Valencia -> San Sebastian -> Barcelona

Valencia, Barcelona and San Sebastian are far apart. If you are set on heading north, I would personally recommend you visit Bilbao instead of Valencia.
But if you want to combine the three cities you mention, then that Volotea flight is by far the best option. Any other option would take a whole day.
San Sebastian to Barcelona has the option of a direct train taking about 6 hours, or you could fly again.

And the best option for Barcelona to Valencia is the train.

Thank you for the response!

I'm seeing that I can get a reasonable rate on rental cars through Costco. We can rent from Valencia airport and return to Barcelona airport. A 6-7 hour drive does not really phase us. We're used to taking 11-13 hour road trips. Our lodging in SS offers free parking. But we don't really know what it's like to drive in Spain. Are we setting ourselves up for some trouble if we opt to drive?

I haven't driven in Europe, but I know one thing you need to be really, really aware of is the possibility of theft from autos when you stop for a meal or sightseeing and have all your possessions with you. Car break-ins are a real risk. At a minimum, you must leave nothing visible inside the car, and you must not open the trunk after you reach the parking spot, because someone may be watching to see whether you have interesting things stored there. I would have a tote or backpack in which I placed expensive electronics, prescription medications, etc., and I'd take that with me wherever I went. I'd have my excess cash, back-up credit cards and passport in a money belt, neck wallet, or some other storage spot under my clothes.

The car will not be useful for city sightseeing, and parking may not be simple. Choose lodging partially based on availability of parking.

You might want to shop around for car rentals. My go to is Auto Europe, They have consistently had better rates than Costco. It also doesn't hurt to check the various rental companies(Hertz, Avis etc), sometimes you can get a pleasant surprise
It would be much more efficient to fly into or out of Bilbao. Brad

I'd MUCH rather fly from Valencia to Bilbao or S Sebastián than drive across half of Spain in a day. No contest.

I'm so torn! We really want to go to both Valencia and San Sebastien.

But if we were to keep Valencia and take out San Sebastian, I suppose we could go to Madrid after Valencia. It's the heat that is turning us off from Madrid. My 76-year old mother with low blood pressure can't function in extreme heat. Any other suggestions for where we might be able to visit as a 2nd city in our 3-city tour of Spain? Valencia, another city, and then ending in Barcelona.

Thank you all for your suggestions!

You could stay in Girona for 2 nights. It's very charming, close to Barcelona, and you could do a few hours' trip to the coast or to Besalu.