The Prado Museum is the first must see. All of the famous Spanish painters like El Greco to other European masters (lots of Rubins). There are English-speaking tours but we chose to wander on our own. Guards were very helpful pointing us in the right direction so that we did not miss anything that we wanted to see.
The Presidential Palace is on the other side of Madrid but easily reached on foot. Again, we went by ourselves rather than in one of the group tours but we occasionally stopped by a group to hear what the tour guide was saying. The Palace is breathtaking and well worth the walk.
We ate and drank at the Plaza Real where you can sit at one of dozens of cafes and eat tapas and drink a cerveza (beer), vino (wine), or agua con gas (sparkling water). The Plaza Real is centrally located and the best place in the city to people-watch and soak up local culture.
Quick Tips:
Best Way To Get Around:
Walking is the best way to get around. The Grand Via bisects the city between the Prado Museum and the Presidential Palace. It, and streets close by, has lots of hotels. You can walk to any of the major sites in the city from this area. Traffic is a bear and the few times we did take a taxi (to train station and airport) it was an adventure.