1) Palacio Real (Metro stop: Opera, Line "R" or Plaza de Espana, Line 10). The former Royal Palace is one of Madrid's major sites, and is very close to the site of the modern-day Senate. There are citywide bus tours offered by different companies (you will have to register for them) that start from the Palacio Real. There is also a garden with statues of famous Spanish rulers situated around the perimeter, so even if you do not go on the bus tour, you can appreciate a stroll around the neatly trimmed greenery.
2) Parque del Buen Retiro (Metro stop: Retiro, Line 2). This large park, located in the middle of Madrid, attracts thousands of people every day. It's a perfect place just to relax and take in scenery. There are running trails for people who want to get in a morning workout, monuments as well as statues of historical or artistic importance for history buffs and public gardens for nature lovers. Visitors can rent boats and take them out on a lake in the center of the park. Artisans and "street" artists line the walkways of Retiro, selling their creations. One of the artists I saw did "splash art" with spraypaint, and, only using the paint and different cut-out shapes of paper, managed to create paintings that look like they were done with oil paint on a canvas. These "starving" artists certainly have talent, and the average cost of a painting was 10 euros. Even if you don't buy anything, it's still fascinating to watch the process of how a painting is made.
3) The Madrid Zoo (Metro Stop: Batan, Line 10). Located in the Casa de Campo, this zoo (which I blogged about earlier) has almost every type of animal imaginable. It also has an aquarium with live dolphin and sea lion shows and attractions for the kids. For those Rhode Islanders reading this blog, I would say it is like a mix between Sea World and Roger Williams Park Zoo. Definitely something to do on a "low-key" day.
4) Plaza Mayor (Metro stop: Sol, Line 3). What happens in Plaza Mayor varies from day to day. From having bull fights in a specially-constructed ring (usually imported from Navarra or the north of Spain, as outside of that region, bullfighting is not popular in Spain) to horse shows to fashion shows or displays of artwork, there is usually some new project being constructed for some show at Plaza Mayor. However, even on days when the square is bare, it's a beautiful piece of architecture and the perfect place for an afternoon stroll.
For food, Madrid has any number of restaurants. Although it's a bit pricey, La Paella Real, next to El Palacio Real, is a wonderful restaurant that's well worth the cost. Plus, the traveler gets to sample the customary Spanish dish: paella, or rice with vegetables and your choice of meat. Be warned--to show the freshness of the food, Spanish restaurants serve seafood like shrimp with the heads still attached! You'll just have to shell the shrimp yourself.
VIPS, which is like a mini-CVS, is the store where you can buy almost everything. It also has a VIPS restaurant, which serves burgers, chicken and more "American"-style food. The VIPS on Calle Velazquez (Metro: Avenida de America), near my school, also had two additional restaurants, including an Italian eatery, in the basement. The food is reasonably priced at all VIPS establishments.
And if you don't feel like that, you can always avail yourself of one of the many tapas bars around Madrid. Usually set up like small taverns, these bars serve alcoholic and non-alcoholic beverages and tapas, or reasonably-priced appetizers ranging from Spanish omelette (potatoes and eggs) to jamon serrano (pieces of cured ham). Most of the places will have a sign outside advertising "tapas."
For a good dessert or snack, go to the Chocolateria San Gines, located in the alley off of Calle Arenal next to the Church of San Gines, where you can order churros, or long strips of fried bread, with chocolate. The chocolate comes in cups like hot chocolate, but has the consistency of a thick pudding.

Try them--they're delicious! And you can also justify indulging yourself, because you'll burn the calories by just walking from place to place through Madrid.
For souvenirs, there are many stores around Sol with good-quality, cheap goods. You can also buy souvenirs at pricier stores in the areas of Serrano and Colon, if you want. Some people choose to buy items off of street vendors, and you can--but be aware that not all vendors are the same, and some will try to cheat you, especially if you stand out as a tourist.
For nightlife, the bars and clubs tend to be concentrated in Sol or along Gran Via, a major artery in Madrid. The neighborhoods of Alonso Martinez and Chueca also have great club life and bars ranging from "hole-in-the-wall" places to posh Manhattan-style lounges. Kapital, located in Atocha, is a popular seven-story club with a different style of music on each floor. It can be pricey, but is a guaranteed night of fun if you do go. On Sunday, while you recover from staying out late, you can sleep in, or, if you're an early bird, you can head over to El Rastro, a huge flea market where almost everything from clothes to snack food is on sale. The streets around La Latina and the Plaza de Cascorro, where the vendors set up their stalls, are packed every week.
If you do go to Madrid, don't stress about trying to see and do everything. The attitude of the average Spaniard is much more relaxed than that of most Americans. If you don't see something or get somewhere, there is plenty to do to occupy your time, and you can always re-attempt your endeavor at a later time. Above all, enjoy! As they say in Spain, "no pasa nada."