I've spent a lot of time in Spain over the years, and I love all of it.
One driving trip that would hit a lot of great cities would be to go to Pamplona, then to San Sebastian and Bilbao is right next door. Pamplona is where they run the bulls, a classic old Spanish city, and San Sebastian is a beautiful Basque coastal city - originally a big fishing port. I remember Bilbao as a very gritty place, but that was decades ago and I'm sure it's very different now. From Bilbao, you can head west to Santander which is a very old university town on the coast with beautiful architecture; or, if you choose, turn around and head north from San Sebastian to go back to Barcelona through France. With that route you have the opportunity to see Lourdes, and also Carcassonne, two very interesting places and head back down the Mediterranean coast.
The other option is to go to Madrid via high speed train, and take day trips to Segovia, Toledo and Avila. You can do Segovia or Toledo, drive to Avila for the night and continue to the other city for the trifecta. Segovia is a must-see if you're in the area. Toledo is the next most popular destination, and Avila is probably the least traveled and there is not a lot to see, but it is a incredibly well-preserved example of day-to-day living in a walled city and I always find it interesting to stop there. I recommend staying at the Parador in Avila for a very authentic and lesser traveled walled city experience.
All the Paradores are amazing, relatively inexpensive, and usually the coolest hotels in the city. It's fun just to view their properties
online, and it may give you some inspiration. (I could plan a whole Spain vacation just around their hotels) They are mostly historical hotels that are supported by the Spanish government.
If it's not during the most crowded season, the highway up the coast from Barcelona puts you into Coast Brava near a bunch of really interesting coastal towns like Tossa del Mar and Lloret del Mar. If you go to San Sebastian and head back via France, you can hit them along with Girona heading south on your way back to Barcelona. IIRC, Montserrat and the Dali Museum are also both in that general area of the north-east coastal quadrant of Spain.
Just remember that in Spain they still practice the
siesta, and in many cities, like Segovia, everything will shut down for a few hours in the middle of the day (2-5PM). It's not hard to get shut out of lunch if your activities keep you too busy. ;-) There's usually a bar somewhere that stays open if you get desperate. On the flip side, they party all night - and I mean
all night.
Edit: Re: pickpockets, most of them are Gypsies and they focus on public transportation and high-traffic tourist areas - Esp. Las Ramblas in Barcelona and El Rastro in Madrid. Most you won't ever see, but if you are approached by someone asking for money - usually an older woman - with a bunch of small children, just move away quickly and don't let them swarm you. The kids are the thieves. You're not in any physical danger, they just want your stuff. Don't worry about offending them, they will complain and back off. I usually will wear a
neck wallet in cities where that's a problem. It's kind of an overblown problem, but it exists for sure and is worth taking precautions. Frankly, poorly attended phones, cameras, handbags and luggage seem to disappear a lot more often.
