I enjoyed Burgos a lot. It gets many pilgrims so the pretty historic area is very lively. There's a pleasant walk along the river (as Rick says, the city is very windy). There are two nice convents/monasteries/whatever outside town. Some bus transportation is available, but possibly not every day of the week. The Cartuja de Miraflores was the one I liked best. It has an absolutely stunning Annunciation painted by Berruguete plus other religious treasures. I'm not at all religious but found the Cartuja a very worthwhile stop.
Another possibility if you're willing to rent a car would be the dramatic Picos de Europa area of Cantabria/Castilla y Leon. It's different enough from the beautiful but less dramatic Basque countryside to justify the time, but I probably would give it a stronger recommendation if you were not going to see so much of the Basque country. Potes is a really oretty old stone town.
Laguardia is a very picturesque medieval hill town near the La Rioja wine country. It has some wine bars but you don't need to like wine to enjoy the place. It's possible but challenging to get there by public transportation. It's on the Vitoria-Gasteiz / Logrono bus line. Doubt that this would be more than a one-night stop, though.
Zaragoza is very different from both Madrid and the Basque Country. I liked Teruel even more, but from the transportation standpoint, it is probably a bridge too far.
Salamanca has a huge historic district and could be a 2-night stay.
Toledo is fabulous, and I was glad to have three nights there. Others are put off by how quiet the city gets when all the day-trippers have left, but I have usually crashed by then, anyway.
Cuenca (inexpensive AVE trip if tickets bought early enough) could be a day trip from Madrid or an overnight stay. Very nice medieval distict and a couple of good modern-art museums. You'd need to take a bus or a taxi from the AVE station into town, and I think buses are infrequent on weekends.
I imagine Segovia is already on your daytrip list, but I thought it was worth an overnight stay.
Two less common destinations I enjoyed visiting from Madrid are Alcala de Henares, a very pretty university town accessible via local train, and Chinchon, a small historic town reachable by bus.
