The Zona Rosa is quite central for the Museum, Zoo, Frida Kahlo's place, the 'Castle' and the silver market. Just be careful because Mexico City can by quite dangerous. Having said that, I do love it and I love the Mexican people.
Get the hop on hop off bus that goes around the city. Most hotels have day trips that you can do (via a local company). I never do organised trips as a rule but the ones in Mexico City are really good value. You can do the pyramids, Taxco and lots of other places for only about £30 or £40 or so.