There's a lot of things to do/see/eat/taste in México city, but definitely go to National University main campus (Ciudad Universitaria a.k.a. "C.U.") A place most travelers skip is the "Postal palace" this is a building in the historical center of the city wich is gorgeous and incredible beautiful inside, despite the exterior may look another regular building, the inside will let you jaw drop! As for the food, try going to "Mercado de San Juan" Local market where you can find anything you may imagine from exotic fruits to lion meat and all the gourmet things you can possible dream. Recently the Soumaya museum opened with its amazing futuristic building and the largest Rodain sculptures collection, this museum is for free!! Anthropology museum is a classic and a must, but i will also add Modern Art Museum and it's "Art trough the history of México as a Nation" permanent exhibition will help you understand a little bit more about our history and culture.

Museum of Anthropology would be first on my list (watch the voladores outside and walk down to the Tamayo museum just down the street), Bellas Artes, Secretariat of Educacion for Rivera murals, and the pyramids at teotihuacan are spectacular. Getting there and back will fill a full day. And Fridas Blue house in Coyoacan is great if you are an art aficionado. In the evening Garibaldi square is great - filled with roving mariachis. Take taxis or get a car and driver from your hotel or a taxi stand or take the subway (easy to get around) - don't hail a taxi on the street in Mexico City.

Frida kahlos and diego riveras house, the house where trotsky lived, take a bus tour of thecity, the red ones hop onn and off , this stops starts by the zocalo, cathedral. To those who suggested to visit chichen itza, it is a bit too far from mexico city, more than 1000 km, i know cause i live in the yucatan.

Chichen Itza the world of the Mayas ! Interesting and lots of history. Amazing structural art!

go see the pyramids of Teotihuacan

Visit the museums like modern art, Bellas artes,theres lots these are just some just to say, and the zocalo.