As everyone else says, Granada is amazing. The Alhambra, well, words fail.

Seville has some pretty cool stuff too and is maybe two hours or so from Granada by bus. However if you go there after Granada, the alcazaba there is not going to seem as amazing—even though it is.

I only passed through Jerez on my way from Cadiz to Seville. It looks like it could merit a stop. I really enjoyed Cadiz, but it may not be what everyone wants to do on a major trip.

Practical matters: Granada is about 4 hours south of Madrid by bus. There are tons of buses going from one to the other. Make sure you get a ticket to the Alhambra in advance. It’s not as crowded in late November (when I was there with my wife) and the weather is nice (well, better than Chicago), but get your tickets in advance. I didn’t find many people in Granada/Seville that spoke English too well, but I did make do.