In terms of Aztec history , there are a lot of must sees :
El Templo Mayor : Its in the Zocalo , city centre and is the ruins of the great pyramid of Tenochtitlan , the most famous of all the Aztec pyramids, and has a superb museum.
The serpent head cornerstone : I wrote a post about it on AO, its also in the city centre and worth seeing , its basically part of the museum of the city of Mexico but it once stood in the Templo Mayor. Its interesting to see because it just shows how important and symbolic snakes were for this civilization.
The monolith of Tlaloc : : I also wrote a post about this statue of the rain god for AO, its located in Chapultepec park and he stands directly outside of the Museum of Anthropology.
The Museum of Anthropology : This museum is definitely a must see , it has a huge room displaying artifacts of the Aztecs but it also has rooms for other Meso-American civilizations such as the Mayans , the Teutihuacanos, the Toltecs, the golf coast Olmecs , the Zapotecs and Mixtecs, and the cultures of the arid North.
There is also a very significant pre-Aztec ruin to be seen in DF , the Pyramid of Cuilcuilco which is the oldest of all the Americas.
In terms of advice , I would say be alert to your surroundings in the city , it is unfortunately getting more and more dangerous every year as the violence from the surrounding states reaches the capital.
A lot of this is going to be just generic staying safe advice but :
Try not to carry your valuables in your pocket (especially not on public transport) , dress normally as dressing up can make you a target for crime
If you speak Spanish , that is brilliant , try to speak it as much as often to avoid appearing too “Gringa” which may make you appear like a target for unwelcome male advances and miraculous rises in prices when buying things etc.
Don’t take taxis that don’t have a licence and if the taxi meter is switched off while driving tell the driver who may have conveniently “not noticed” that you would like the meter to be switched on because you wont pay any “gringa / guerrita” tax.
If you feel that you are being followed go to a public area immiediately and speak with the police or any authorities available.
Do not go to “Barrios bravos” such as Tepito, Iztapalapa , Ecatapec , Santa Fe , La Merced etc. Just dont. Yes , they may have interesting Santa Muerte shrines but you risk ending up actually meeting La Santa in the afterlife which could well happen in those kinds of areas where violent crime is just a cotidian reality.
If you are mugged (Its unlikely it will happen so dont be scared ) , don’t fight or make any sudden movements ( have slow and steady hand movements , avoid eye contact) the robber just wants your money /phone/ valuables etc. often these guys are high on crack cocaine and if you put up a fight you could end up dead , prioritize your life over any material possesions.
In terms of food , I’ll write another comment with some reccomendations in a bit