If you only had one day to spend in Mexico City then a great place to spend it would be in the Zocalo, or main plaza. The Zocalo is at the heart of Mexico City. It would be easy to spend an entire day with just the sights on offer here, but I would allow at least a half day at the minimum.
The three main sights to take in here are the National Palace, Metropolitan Cathedral and the Aztec ruins of Templo Mayor. Depending on your pace and inclination each of these sites can be visited in one half hour to 2.5 hours for the detail-oriented visit. I personally spent a half-hour at the Cathedral, one hour at the Palace and about 2 hours at the Aztec ruins.
The Aztec ruins of Templo Mayor were destroyed by the Spanish and were not excavated until the late 1970s. In Aztec belief Templo Mayor was built in the center of the universe. The temple was enlarged several times and you can see the layers in the ruins today. The Spanish used the stone from the temple to build the Cathedral that stands next to the ruins today. The models in the attached museum give you an idea of how impressive they were before they were destroyed by the Spanish.
If you are traveling on a limited time frame and you do finish with the Zocalo attractions within a half day you have several options for the rest of the day. The National Museum Of Art and The Palace Of Fine Arts are a 20 minute walk away from the Zocalo. A 5 minute walk from these museums is the Torre Latinoamericana (60 Pesos fee) for an over 40 stories-up view of the Capital or Alameda Central if you just want a pleasant stroll through a park. In addition, Plaza Garibaldi and the mariachi bands await you after a 20-30 minute walk from the Zocalo as well.
Mexico City is massive and hectic and it can tend to overwhelm you. So if your limited on time or if you just want to “get your feet wet”, spend your first day in the heart of Mexico City, the Zocalo. Limiting your activities to this area can lower the stress involved with touring Mexico City. You probably won’t have to deal with riding the Metro or buses and taxis. This will allow you to acclimate and get your feet on the ground. Do yourself a favor and allow Mexico City to penetrate your senses by careful contemplation of its heart before you launch into your full-blown tourist assault on its entirety.