Madrid has a big metro system and it is really easy to get around. Pretty much everything you’re going to want to see in the center isn’t very far from each other. We stayed with someone on couchsurfing that lived close to the main center. Go to a main tourist office and get a map of the center that has all the main things you’ll want to see.
When you walk around the main center of madrid near Puerto del sol, there are tons of little bars you can get cheap beer, jugs of sangria, and tapas. Some of them will serve you free tapas every time you order a round of drinks. You also have to try a churro with chocolate at a churreria.
Check out the Temple de Debod, go at night when it’s lit up.
Madrid has some really good markets selling small portions of all different things so you can try a different variety of food and drinks. Go to the Mercado de San Miguel and Mercado de la Cebada.
Walk around the parque de el retiro and check out the Alfonso XII monument in the lake and the crystal palace that’s inside the park.
We went to a really cool museum that is free from 6-8pm called Museo del Prado. They have beautiful paintings and some work by Picasso.
We went to the Fiesta de la Paloma in la Latina which was going on the whole week that we were there. It goes on the middle of August for a week but gets really big on the weekend. They had live music, beer and food trucks everywhere, and carnival games. La Latina is a good area to go out at night with streets filled with different bars.