If you really want to get off the tourist trail in Madrid, there are two places you HAVE to visit.
1. Lhardy, a restaurant/shop that has been in business since 1839. You can try their famous caldo -- beef consommé, and enjoy a sweet or savory pastry (I had a savory pastry with some sort of spicy sausage filling... Tasty!!).
2. Casa Labra -- a restaurant/tavern that has been in business since 1960. If you hit them up at lunch time, you can try their tapas de bacalao -- a piece of codfish that is battered and fried, or their croqueta de bacalao -- a codfish croquette. Both are delicious! To wash it down, try a tiny glass of vermouth (rojo, not blanco). However, keep in mind that there's often a line going out into the street (check out the pic!), and it's standing room only once you get inside. It's worth it, though!