It is a good idea to devote your first day to a circular walk around the historic center of Madrid, which will allow you to see the most iconic sights and visit at least one museum of the Museum Triangle of Madrid.
Start at Puerta del Sol, then, after a good look around, follow Calle Arenal west to see the magnificent Royal Palace of Madrid and the nearby Almudena Cathedral - the main religious temple of the city. You should definitely visit both at least for a short time!
After that, follow Calle Mayor east towards Plaza Mayor, arguably the most beautiful square in Europe (some Parisian squares may disagree) and most definitely an architecturally perfect creation. Take Calle Atocha from there which will lead you to Museo Reina Sofia - one of three museums that make up the Museum Triangle in Madrid. If you prefer to start with a visit to Museo del Prado, turn left on Paseo del Prado and go explore the world's largest art gallery first!
After Museo del Prado continue up Paseo del Prado to Plaza Cibeles, a square famous for its fountain sculptures and football celebrations courtesy of Real Madrid FC. Follow Calle AlcalГЎ towards Puerta de AlcalГЎ, a famous Madrid sight akin to Arc de Triomphe in Paris and Barcelona. After all the photos you took, come back along Calle AlcalГЎ to hit Gran Via, Madrid's most central avenue, where you will probably start looking for a place to eat or go back to your apartment for some rest before supper. If you are not completely exhausted, follow Gran Via to Plaza EspaГ±a and head to the Monument of Debod - an ancient Egyptian ensemble right in the centre of the modern Spanish city!
You should devote your second day in Madrid to visiting Museo Thyssen Bornemisza and Museo Reina Sofia, if you didn't make it there on the first day, and thus get a feel why Madrid gives Paris a run for its money when it comes to fine arts. After museo Thyssen Bornemisza you are only steps away from the Buen Retiro park, which you could explore at your leasure for good several hours. Still haven't gotten enough art? After the walk in the park, you can pay a visit to Caixa Forum Madrid, an events and exhibition space sponsored by a Catalan bank. The entrance is always free and art exhibitions held are varied and excellent. Plus, you will end up right on Calle Atocha, a very popular street with many drinking and dining options.
Map of this itinerary