Another really cool thing close to the Zocalo is the Templo Mayor. Its a super old temple that the Spanish colonizers built over when they settled in Mexico City. In the 1970's it was uncovered, so now there are literally ancient ruins sitting right in the middle of the city. You can walk up and look at it through a fence or you can pay around $4 to go walk around inside and go to the museum.
There are layers and layers of temples, as each new king expanded the previous temple to symbolize the growth of the empire.