Even if you're not particularly an art person, it's worth checking out Madrid's museums because they're free (or at least during certain times; generally the last couple hours before closing). The cavernous galleries are the perfect escape from the scorching heat (except for the part where you have to stand outside in the sun; bring a bottle of water), and because the trifecta of art museums are all quite close to one another, well, there's no excuse not to hit up at least two of them.
The Reina Sofia houses the modern art collection in a sleek building that is aesthetically pleasing but a practical pain in the ass. (I gave up trying to reach the top floor because I couldn't find an elevator that would take me and there was nary a staircase in sight). But it has the distinction of owning Picasso's Guernica, which is a must-see. Guernica is essentially the Mona Lisa of the Reina Sofia, but better because the painting is so large that you can still see it even when the gallery is packed with people.
El Prado has all the hoity-toity classical stuff, if you're into that sort of thing. I'm a fan of Goya, myself, but after endless wandering everything sort of started to look the same so I called it quits and left after about an hour. The real treasure, in my opinion, is the Thyssen, which as a former private collection has quite the eclectic mix of artists and styles - essentially a little bit from every major artistic movement from classic to modern with everything in between.
Two of my friends told me I had to visit the Mercado San Miguel, which I did. Several times. Located right by Plaza Mayor, this vintage industrial building is a market in the same way that Chelsea Market in NYC or the Ferry Building in San Francisco are also "markets." By which I mean that they're less about fresh produce sold by the farmer/butcher/fisherman's family and more up on trendy artisanal pre-made or cooked-to-order dishes. It's not a bad thing, per se, but very bougie.
El Rastro, just a couple streets over from where I stayed in La Latina, was a whole other experience. Every Sunday, it takes over the neighborhood and becomes a giant, sprawling open-air flea market. For the most part vendors are hawking standard items like clothing, jewelry, antiques, and artwork, but I did see a plant stall featuring venus fly-traps.
Since I managed to refrain from spending much - or any, really - money at these places, I decided to reward myself on the last day with a tour of Real Madrid's stadium. But here the story takes a little tragic turn. Earlier in the afternoon I had dropped my camera while readjusting (wide angle lens are heavy, so I periodically switch which shoulder I have it slung over). It wasn't from a very great height, so at the time I just checked to make sure it could still turn on. (It did). But when I pulled it out later to take a picture of the soccer pitch from above, all I could see through the viewfinder were little broken bits of whatever it is that makes the camera body work. So...iPhone photos ahead.
The tour leads you up, down, and all around the stadium with I suppose the major highlights being the trophy room (some serious metal in there; kinda reminded me of Hogwarts), sideline (you can sit where the players do during games) and changing room.
Sports fans already know this but Real Madrid has an incredibly rich history and reputation. There's also seems to be a strong sense of national pride as almost all the players are Spanish (as compared to the English premiere league teams that are predominantly comprised of internationals). Definitely worth it if you're a football fan, and maybe even if you're not - I'm glad I went!