What is your favorite city or town near Madrid?

I loved Salamanca, Leon and Segovia, I liked Avila. I would easily go back to and spend a few days in any of the first three. I also really liked Burgos.

Here's my photos - https://andiamo.zenfolio.com/p898177329 and https://andiamo.zenfolio.com/p551756686

Segovia and some would probably recommend Cuenca.

You could take the AVE high speed train to the charming cliff-side town of Cuenca, which is about 1 hr east of Madrid. Cuenca, a UNESCO World Heritage Site, is a 12th century medieval town built on the steep side of a mountain, which makes for a very unique place to spend a day or two. The forested hills nearby also make for some nice light hiking! :)

Salamanca definitly. Great mid-size town, probably the nicest plaza major in Spain, great architecture.
Avila didn’t cut it for me.
Segovia has a few impressive sites, such as the palace and the aqueduc, the main church as well.
But I’d say there’s more pure beauty in Salamanca.
I also absolutly love Toledo. So much I went twice. :-)

Avila comes way down my list. Aside from Toledo, I really liked Segovia, Cuenca (buy that AVE ticket early or it will be $$$) and Alcala de Henares. The latter has frequent commuter rail service from Madrid even on Sunday. Cuenca is very much a full-day's sightseeing if you like modern art, because it has two nice, small museums. Segovia is more than the acqueduct. Salamanca has quite a large historic district (peep into the open courtyards you come across) but is considerably farther than the others.

The little town of Chinchon (that's a bus trip from Madrid) is interesting for perhaps a couple of hours. The others I've mentioned--with the possible exception of Avila--have more to recommend them.

Also, a great little day trip from Cuenca is La Ciudad Encantada, about 30 km away. La Ciudad Encantada is a protected geological site with many large and unusual rock formations created through the erosion of the Júcar river, which makes it look like an "Enchanted City", which is it's namesake in Spanish. Many films were shot here, including Conan the Barbarian with Arnold Schwarzenegger. The entrance fee is only 4 €!

The only town I can think of not already mentioned is Zamora. Seems you’ve got plenty of places to choose from.

I really like Alcala de Henares....a small university town about 50 minutes from Atocha on the Cercanias. From there you could also go out to Guadalajara. If outdoor activity is of interest to you, you could go out to Manzanares el Real and go hiking at La Pedriza.

I second all that people have written about Salamanca (perhaps my favorite) and Cuenca. (If you take the AVE to Cuenca be aware that the AVE station is quite far from the city....probably 20-25 min in taxi) There are busses but you have to factor the bus travel time and changes into your schedule) As an aside if you go out to Ciudad Encantada you can go hiking there or you can go horseback riding or kayaking. If you are up to traveling that far, Leon would also be a wonderful destination. Lots to see and do.

Within an hour or two of Madrid there are tons of interesting places to visit.

Within your three hour train journey you can get quite a distance if you're using the high speed services. Zamora has been mentioned (and I'd agree with considering that), but in the other direction you could reach Valencia (less than two hours in fact). I'm not particularly recommending Valencia, mind, although it does meet your preference for walking/biking around for a couple of days rather than intense tourist visits. There's nothing wrong with anywhere mentioned here, and all would be great, based on your stated interest in enjoying the local "vibe" rather than sight-seeing, I'd suggest somewhere such as Zamora, Avila and, particularly, Alcala de Henares rather than the guide-book big hitters.

Here is a suggestion that I don't think anyone else has mentioned: Sigüenza. It is about a 90 minute train ride. I like towns over cities and this old town would be a great place to wander and explore. Do some research on it to see if it appeals to you, complete with castle, as well as Roman, Visigoth, Arab and Christian influences. All of your other suggestions are also enticing and you will have many great options.