This was my first visit to Mexico. Since I've been here, I've been surprised by the way everything is used to its maximum potential (i.e. nothing is put to waste - I say this as a typical American). The metro is almost always completely full, every inch of street and sidewalk is used by vendors, the buses are packed, and there seems to be a deep, unbending, patriotic ferver that runs through each of the calles and barrios.
Quick Tips:
A week´s worth of knowledge in just a few bullet points:
1) Spanish is not necessary in and around Mexico City.
2) Check out the Zolaco (main square, metro at Zolaco) at night. There are generally some indigenous dances occuring to celebrate that period of the ancient calendar.
3) Anything you could possibly want to buy (i.e., food, clothing, crafts, dishwashers, etc.) is available on the street. I wouldn´t buy anything in the store unless you want to pay more for it. Bargaining is possible as well, but look for the signs with prices marked.
4) Almost everything is closed on Mondays, so this is a good day to travel to a tourist site, such as Teotihuacan or Cholula (both pyramid sites).
5) The pyramids are worth it, especially Teotihuacan (take a full day for it, similar to Tikal in its splendor).
6) Other sites that were particularly memorable for me included the Museo de Anthropologia, the Museo de Belles Artes, the top of the Torre Latinoamericano building (second largest skyscaper--can't miss it), Alameda Park, and any of the several markets/panhandlers on the streets (especially around the Zolaco).
Best Way To Get Around:
Metro is the best way get around, period. There are maps and clear signs everywhere. Pick up a ticket (2 pesos) from one booths as you enter. There are nine lines total, and each is SUPER-packed during rush hour (or just about anytime near downtown). Don't be afraid to push people around; they certainly won't mind pushing you around.
Secondly, buses are great for day trips, but a bit more expensive (average US 3-8). Use a lonely planet guide to get the best price, or shop around. We used Estrella Rojo, which was cheap, air conditioned, and had plenty of leg room (they specialize in and around puebla-cholula).
From the airport, its best to take the metro (head towards Salida A and follow the signs). The taxis will only take advantage of you.