Hi David- you guys are in for an awesome trip! We were there last November for 2 weeks, and the weather was cool-to-mild in both Barcelona and Madrid. We also visited farther north, along the coast, and there we got some rain and wind and it was on the verge of cold, but not actual freezing cold. Not sure what you consider "cold," but I'd say plan to have a jacket with you, and be prepared with a sweater or other layer if you need additional insulation.
We didn't experience any crime (see below about trash, which you may not encounter), but use a moneybelt and don't go in a neighborhood if you suspect it's unsafe (we didn't find any such place). As you said, take obvious pecautions you'd take in any big city. Police are around and don't hesitate to ask them for help.
We used our Visa card most of the time, but also got cash from ATM's. Our credit card has a European-style chip in it, but we still have to sign a receipt to complete a transaction. If you have a regular American credit card, with a magnetic strip on the back but no chip, you should be fine -- waiters, shops, hotels, etc. can still swipe you card and have you sign a receipt. Over the next few years, the USA will be converting to the chip-and-P.I.N. credit card system, too.
As you indicated, Granada and Seville are worthy desitnations, but in a country as big as Spain, we've always either done Southern Spain or Northern Spain, but not both, unless we had more than 2 weeks.
Are you taking a fast AVE train from Barcelona to Madrid, or a quick, cheap flight, or bus, or rental car? If you have the time and available transportation, consider Girona, an hour north of Barcelona. Its old town is atmospheric, with winding stairs and narrow passageways, and lots of inviting plazas. Great places for tapas. We took the train and stayed one night on our short trip, but could have easily devoted another full day there.
To the northeast of Madrid is Segovia, a favorite town for at least a day trip, if not staying overnight, with a fantastic ancient Roman aqueduct and other sights. It's also renowned for its roast suckling pig (cochinillo), which we had for dinner one night last November. You can easily get there by bus or train, if you're not driving. Same with Toledo, to the south of Madrid, where the painter El Greco lived and worked.
I'd save Valencia, Granada, and Seville for a Southern Spain trip another time and focus on Barcelona and Madrid, with a possible side-trip to the place I mentioned above. If you enjoy art, Madrid's Prado, Reina Sofia, and Thyssen-Bornemisza are all world-class.
BTW, in November last year, there was a Madrid garbage-collector's strike, and trash was piled up all over town, inluding green bags people used to pick up after their dogs. The green bags littered the sidewalks and many were split open. That was nasty, so hopefully you won't encounter that this time!