Madrid's most valuable art treasures are all on display within a few blocks of Paseo del Prado. This area is home to the Prado, with its astounding selection of masterworks from Velázquez, Goya, El Greco, and others; the Centro de Arte Reina Sofía, with an excellent collection of contemporary art; and the Thyssen Museum, 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 Museum).
Any visit to Madrid should include a walk in the old area between Puerta del Sol and the Royal Palace. 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. End or begin the day visiting the Royal Palace, built some centuries later than many of the churches but nonetheless quite staggering.