'''Saturday morning''': start at the '''Plaza de Oriente''', the majestic plaza between the '''Royal Palace''' and the Opera house (metro: Opera). Wander south along calle Bailen, toward the '''Cathedral of Almudena'''. Turn left (away from the cathedral) onto Calle Mayor. Walk along calle mayor, until you pass a small plaza on your right. This is Plaza de la Villa, where the Casa de la Villa (el Ayuntamiento), the seat of the municipal government, sits. The building was built between 1644 and 1696 and designed by Juan Gomez de Mora, the architect of the Plaza Mayor. The statue in the middle of the square is of Alvaro de Bazan, the Admiral who planned the Armada.
Continue along Calle Mayor, until you reach the arched entrance to the '''Plaza Mayor''', on your right. Enter the Plaza. Originally concieved in 1581, under the reign of Philip II, construction wasn't started until 1617. A series of fires badly damaged the plaza, so it was reconstructed in 1790 architect Juan de Villanueva. This is the version we can see today. The statue in the center is of Philip III on horseback done in 1616 by Juan de Bolonia. The Plaza Mayor has hosted markets, bullfights, soccer games, and public executions.
Return to the Calle Mayor, and continue to the '''Puerta del Sol'''. wander through SOL, and take Calle Alcala to leave. Wander along this street until you get to '''Plaza Cibeles'''. At this point The Palace of Communication (Post Office's main branch) is in front of you on your right, the Casa de America is in front of you on your left. Immediately on your left is an Army building, surrounded by gardens, and immediately on your right is Banco de Espana.
Turn right onto Paseo del Prado. Wander south, passing the Bolsa de Madrid (stock market), with the Tomb of the Unknown Soldier and eternal flame in front of it.
'''Saturday Afternoon''': At 2:30 on Saturday afternoons, the '''Reina Sofia Museum''' of art stops charging admission fees. It is easily worth 2 to 4 hours of your time.
If you still have energy after all this, walk north along Paseo del Prado to Plaza de Colon (or take bus number 14) and go see the '''Archeological Museum.'''
In the evening, have thrifty dinner of montaditos or splurge at a nice restaurant.
Go to the '''Prado Museum''', and after stroll around the '''Retiro'''.