As mentionned above: the Prado and Centro de Arte Reina Sofia. The first one is classical paintings, the second is modern art and contains Picasso’s Guernica. Lose yourself in the narrow small streets south of Plaza Mayor and have tapas there. Avoid Plaza Mayor and Puerta del Sol for food as it’s too touristy but do have a look at them for their architectural value. Then the statue of Cibeles. Have a stroll through Parque del Retiro. If you feel like it, go to Atocha station to see the memorial to the victims of the 11-M bombings and to the Bosque del Recuerdo (inside Parque del Retiro) dedicated to the same thing. The Royal Palace is great too. And if you still have time, do a day trip to El Escorial. If you can rent your own car and have a good map, the sierra between Madrid and El Escorial is beautiful.