Answer:
The same things Americans do, depending on money:
People who stay home usually watch TV (there is satellite TV as well as cable), or listen to the many varieties of Mexican music, such as norteña, cumbia or salsa. Pop and foreign music are also popular, but on a lesser degree and mostly listened to on larger cities. People who have a higher purchasing power and who own a DVD or Blueray, rent movies to watch them. Those who own a PC and an Internet connection, surf the Web.
Youngsters usually play videogames (Xbox and PlayStation are popular) but if their parents don't have money for such amenities, they usually hang out with friends, attend parties or play outdoor sports (see related questions).
Many people relax on open spaces on their free time, such as public parks an plazas. For instance, the Chapultepec Park in Mexico city is very popular, as it has some child playgrounds, lots of green space and the Castle and Museum at the top of the Chapultepec Hill, in the middle of the park. It also has an artificial lake, where many people rent small paddled boats.
Many others go to the local bazaars (Spanish: mercados sobre ruedas) where they can find cheap clothes, small jewelry or things such as toys, music CDs or handcrafts. Other people go to the malls, just as Americans do. There they could buy some stuff, see or rent some movies, and eat at the food courts or at the fancier restaurants within the mall complexes. One example of these is the Angelopolis Mall (Centro Comercial Angelopolis) in the City of Puebla.
On small towns, usually on Sundays at the central plaza or zocalo, it is common to find small fairs (Spanish: ferias) where food and small farm products are sold; amusement rides such as Ferris wheels (ruedas de la fortuna), or bumper cars (carritos chocones) are installed.
People with more money on their pockets could travel to other cities to visit relatives and/or look for the local amenities. They usually travel by car, but bus and airplane are also common. Many people travel across federal roads or highways, just as many Americans do, visiting small towns and eating or staying the night at the local inn (Spanish: posada). This form of traveling is called pueblear.
People with even more money could go to resort cities, such as Cuernavaca, Veracruz or Acapulco, where beaches, pools and the night life are the main attraction; conversely, many people from smaller cities go to the larger ones (Tijuana, Mexico City, Monterrey) for their night life, enjoying bars and clubs. People with higher purchasing power go to more expensive cities such as Los Cabos, Cancun or the Mayan Riviera, and even some leave the country for places like Orlando (Florida), Cuba or Europe.
There are of course many other attractions, which would include theaters, movie theaters, circus, libraries, cafes and many others that are common on any Western country.