The best thing that we did was take a day trip to Teotihuacán and the Basilica. Teotihuacán is the Aztec ruins outside of Mexico City. It is an amazing place in many ways, and climbing the pyramids are quite a treat.
Xochimilco was kinda fun. It is about a half hour away from the city, and you go on a large gondola style boat and go along a river. As you are on the boat, sellers of flowers, corn-on-the-cob with mayo and cheese toppings, drinks/beer and sometimes mariachi bands will play a song for 50 pesos.
Taxco - a small town about a 2 hour ride from Mexico City which has the cheapest silver in all of Mexico. They say that silver sellers in Mexico come to Taxco to buy their silver and sell it elsewhere in Mexico. There were some nice deals here, but even better was the city and the views of the city. Just beautiful. If you go, find a nice spot to eat lunch - high up on the hills.
Quick Tips:
If you go to Teotihuacán and Taxco - bring some sneakers. Taxco has some major steep hills so sneakers are the best for the town. Bring sunscreen to Tautichan - my shoulders got a bit scorched.
They say if you are going to get money out of an ATM - don't do it on the street. Find one in your hotel or in a shopping mall.
We were slightly careful with the food - usually I didn't feel fantastic after eating outside of the hotel - but that is Mexico. You gotta love it. I just tried to make sure everything was cooked all the way through - so it was okay. If you order soda, it always comes with ice - so if you are worried about the water, ask for no ice. You can purchase bottled water almost anywhere so there was no issue with water.
No one wears shorts in the city - most people wear long pants or skirts. Everyone told me that Mexicans consider this beachwear, so try to stay away from this.
Best Way To Get Around:
We walked to many places, but if you have trouble with smog, Mexico City has much of it. Since it is at a high altitude, and with the smog, sometimes it is tough to walk around a lot.
We took taxis when needed. Make sure to go into a registered taxi if you use them. We choose to take a taxi from in front of our hotel or other hotels so we didn't take the wrong ones.
We used tours when we could - made that easier.
Driving - there is a ton of traffic during the weekdays. If you can avoid it, I would.
I hear the subway is good - we never took it, but I hear it is extremely crowded on weekdays. Weekend for driving or subway seem a good choice.