Madrid's most valuable art treasures are all on display within a few blocks of Paseo del Prado. This area is home to the Museo del Prado, with its astounding selection of masterworks by Diego Velázquez, Francisco de Goya y Lucientes, El Greco, and others; the Centro de Arte Reina Sofía, with an excellent collection of contemporary art; and the Museo Thyssen-Bornemisza, with a singular collection that stretches from the Renaissance to the 21st century. Each can take a number of hours to explore, so it's best to alternate museum visits with less overwhelming attractions. If you're running short on time and want to pack everything in, replenish your energy at any of the tapas bars or restaurants in the Barrio de las Letras (behind the Paseo del Prado, across from the Prado).
Any visit to Madrid should include a walk in the old area between Puerta del Sol and the Palacio Real. Leave the map in your back pocket as you come across the Plaza Mayor, the Plaza de la Villa, and the Plaza de Oriente, and let the streets guide you to some of the oldest churches and convents standing. The Palacio Real makes a good start and end point.