Begin at the Puerta del Sol, the center of Madrid. If you stand with your back to the clock, Calle Arenal is the second street from the far left leaving the plaza: walk down Arenal and turn right into Plaza Celenque. Up on your left, at the corner with Calle Misericordia, is the Convento de las Descalzas Reales. Follow Misericordia and turn left into the charming Plaza de San Martín. At No. 1 is the Fundación Caja Madrid, called Casa de las Alhajas (House of the Jewels). The exhibitions housed here, often done in collaboration with the Thyssen Museum, can be visited for free. Return from there to Calle Arenal, turn right and walk to Plaza Isabel II; then cross the plaza to your right and walk down the short Calle de Arrieta; at the end sits Convento de la Encarnación. Turn left here onto Calle Pavia (off Calle San Quintin) to enter the Plaza de Oriente.
Here you have a choice of going directly to the Palacio Real, to your right, or visiting its gardens (1 km [ 1/2 mi farther on) and/or taking a cable-car ride. For the latter, cross Calle Bailén and walk to the right: from here you can look across the formal Jardines Sabatini to the Casa de Campo park and the Guadarrama Mountains. Walk up Bailén, avoiding the overpass, and turn left down Cuesta de San Vicente, then left into Paseo Virgen del Puerto for the entrance to the gardens and the Campo del Moro. Exit the gardens through the same entrance and get Paseo de la Florida from the roundabout at Príncipe Pío. (Optional: From that roundabout you can take Bus 33 to the Zoo-Aquarium.) Next, walk for about 10 minutes down Paseo de la Florida until you arrive at Ermita de San Antonio de la Florida, the small church with Goya's frescoes. Across from the church you can find Casa Mingo, an inexpensive, traditional, and popular Asturian restaurant that's famous for its roasted chicken and great cider. Walk to the back of the church and get the overpass crossing the railway; you'll find yourself at Parque del Oeste. Once you get to the end of the overpass, turn right and take Calle Francisco y Jacinto Alcántara to the left and up. You'll pass the Escuela Nacional de Cerámica and the Floridablanca cemetery. Forty-three of the people who died in the events of May 2, 1808 are buried in it. Once you get to the top of the street, if you walk east, you'll reach the Rosaleda, the Parque del Oeste's rose garden, which every May and June holds an International Rose contest. Otherwise walk northward and uphill until you reach the Teleférico (cable car) to the Casa de Campo, which grants panoramic views of Madrid. To see the Egyptian Templo de Debod, skirt the park to the right along Calle Rosales.
Opposite the Royal Palace on the Plaza de Oriente is the Teatro Real. Walking down Bailén with the palace on your right, you can enter its huge courtyard and admire the view from atop the escarpment in front of it. Alongside the palace is the Catedral de la Almudena. Walk past the cathedral and turn right onto Calle Mayor: on your left, on Cuesta de la Vega, are the remains of Madrid's Arab Wall. Walk back east up Calle Mayor, crossing Bailén. Turn left onto Calle San Nicolás to see the church of San Nicolás de los Servitas. Return to Calle Mayor and press ahead: on your right you'll see the Plaza de la Villa Madrid's city hall. Farther up Mayor, bear right on Plaza Morenas and enter the Plaza Mayor through the arch. The Andalusian Torre de Oro bar on the left displays gory pictures of bullfights. On the far side, at No. 33, is the restaurant El Soportal, which gives out the plaza's best free tapas with each drink order. (Watch out for inflated prices at the other restaurants, especially if you sit outside.) Exit the plaza to the left of El Soportal and head down Calle de Postas and back to the Puerta del Sol. Proceed up the right side of the square, past the headquarters of the regional government; in winter consider having a traditional caldo (broth) in the charming old shop at the restaurant Lhardy on Carrera de San Jerónimo.