Take the bus! While I lived in Mexico, if I needed to get somewhere quickly (or if it was farther than I could walk), I would take the bus. Taxis will charge a lot of money. I once paid $70 (pesos) to take a taxi to a school that I could ride for $8 (pesos) taking 2 buses. It was a lot quicker to take a taxi (most of the time), but worth it to ride the bus and save money. Yes, it does take a little while to learn the bus system, so if you're only going to be visiting for a short time, it would be worth it to ask someone to help you figure out what exactly to do.
The Metro in Mexico City is also really cheap. You could ride it for around 20 cents and get anywhere in the city! When you're down there, you don't have to pay for connections as long as you don't come back outside. I would advise against riding it late at night, but I never had any problems riding during the years I lived there. If you have on a backpack, put it on your front and carry your purse in front of you. There is a ton of traffic during "peak" hours, so I would avoid that too! I once was riding with my brother (a rather non-Spanish speaker) and it was during the "peak" time. A man behind me decided that he was going to get on the next car that came through, and it was PACKED! He put one arm on either side of me and started pulling me in! Out of fear of being separated, I started yelling, "NO" and he finally stopped. We took the next, less-crowded car that came!