Not cut out to be a culture critic of any sort or a travel writer, but here are some brief recommendations based on my recent travels:
Barcelona, which I've spent substantial time in, is a great place both to go for a few weeks or more if one has the opportunity. Plenty of interesting touristy things to keep you busy, but it's also just a great place to just spend some time "hanging out." Good food and drink, lively atmosphere, and there's also a beach.
Seville is great. Go.
Granada is worth visiting, even though the hotel we stayed in had Fox News for our viewing pleasure. I thought Alhambra was somewhat overrated, but still worth visiting. The Albaicin neighborhood is great for exploring and re-exploring, but don't order the Tortilla Sacromonte until you find out what it is....
Cordoba the town isn't that exciting, but it's worth going to for the Mezquita which is one of the more fascinating things I've seen - massive mosque, originally begun in the 8th century, then continually expanded upon. In the 14th century a Christian cathedral was plunked in the middle of it, though aside from the altars around the edges the original mosque structure was left mostly intact.
Malaga is probably avoidable, unless one needs one more chance to eat fried fish. The new Picasso museum is worth seeing, though not worth an extra trip unless it's the kind of thing you are really into.