Spend a few days in Andalucia.  Visit Sevilla, Cordoba and Granada.  See the catedral and the Alcazar en Sevilla, La Mezquita en Cordoba and La Alhambra en Granada.  Go to a flamenco show (good in Sevilla, great in the Gypsy quarter in Granada).  Spend a day at the beach somewhere along the Costa del Sol and eat at any of the little restaurants right on the beach (have paella made fresh over an open fire, and eat a plate of boquerones fritos (fried anchovies).

In Madrid, do a day trip to Toledo.  Another day trip to El Escorial/Valle de los Caidos/Segovia.  Visit the Prado and the Museo de la Reina Sofia.  Walk around the area by the Puerta del Sol and la Plaza Mayor.  Go to a Real Madrid (or Atletico Madrid) soccer game.

The 3 things you have to do to understand what Spain really is include: a bullfight, flamenco and food.  See a bullfight in Las Ventas in Madrid, flamenco in Andalucia and food pretty much anywhere.  The best food in Spain is not in fancy restaurants, its in small bars that are found all around.