Some good sights in Madrid

Very nice city! Good, easy public transportation.

Well... There are many interesting attractions in the city of Madrid. If you're the museum type, then you would want to visit "El Prado", an art museum with one of the best collections in the world. You could also try "Museo Reina Sofia", a museum based on contemporary art, and also has a sizable collection of Dali's works (and Guernica, Pablo Picasso's masterpiece). Another interesting museum is the "Dress Museum" (Museo del Traje), which showcases different types of clothes and dresses used by Spanish royalty, all the way to modern fashion.
If you don't really like museums, you could try visiting "El Palacio Real", a gigantic palace which used to be the home of Spain's kings and queens. It has a large armory, displaying different types of armor, shields, swords, etc. The main palace is very interesting, and has a room made mostly with porcelain. As in most attractions like this one, you are not allowed to take pictures with flash. Apart from that, I would absolutely recommend this site to any traveler visiting Madrid.
Away from all the bustle of the city, you could visit the "Parque del Buen Retiro", a huge park with a beautiful man-made lake flanked by large fountains on the side. On Sundays the park fills up with people enjoying the outdoors and the sights. This is a great place to relax.
A good place to take pictures and to look around is the "Puerta de Alcala", a big arch near the Parque del Retiro. It was built on the orders of Carlos III, a Spanish king. It is beautifully illuminated at night, so be sure to visit it during this period of time.
You could visit the "Santiago Bernabeu" soccer stadium, home to Real Madrid, one of the most famous soccer clubs in the world.
Another great site to visit is "Plaza Mayor", Madrid's main plaza. It is basically a very large, enclosed courtyard. The buildings that surround it are filled with many restaurants and cafes. A great place to visit!!
In Madrid, there are countless of places to visit and to explore. These are just a handful of the many sites. Enjoy!!