My favorite walk is to a poorer neighborhood south of Sol - Lavapies. Lavapies is always full of life. There is music being played in the streets, and great smells wafting out of the restaurants. This is also the location of our favorite tapas bar. El Boqueron is a small tapas bar - standing room only - where you can get the freshest and most delicious seafood in the city. Anytime I have the opportunity to go, I will.
One of the ponds in Parque del Retiro.
Parque del Retiro Gran Via
Plaza de Oriente
Casa de Campo: an enormous park on the western end of Madrid
Puerta del Sol
But, really the best thing about Madrid is that you can very easily visit some very cool places nearby. Trains and buses will take you to Toledo (the former capital), Segovia (with its huge Roman aqueduct and its beautiful castle), and finally, Avila.