first of all, welcome!
Mexico City have a lot forest in the city, and is the most important cities in the world with forests inside.
you can see in this map the red colour is the urban zone, so are more natural places as you can see in Mexico City.
the point is that you can go to camp, spend time in the rivers, climb some mountains, some volcanos, enjoy the local food in the rural areas, etc as you want, is very possible and all is in the city.
in my case, i am an urban cyclist, then only i can lend you for a bit, a bicycle for some districts (i mean with the ecobici program). https://www.ecobici.df.gob.mx/
also, you can ask for one, in some kiosks in turistic areas with your passport, and they can lend you the bicycle for 3 hours, when you return the bike, they will give you your id.
the sundays, in reforma avenue, have other programas with only 1 ID, other with 2, or others as Parque España you can rent a bicycle for 15 pesos per hour (aprox).
in fact, other female friend for the critical mass rent bikes for 40 or 50 mexican pesos per trip.
anyway, have a good ride !
sorry, 1 comment about the map.
it is only Distrito Federal.
Distrito Federal have 8 millions of persons aprox.
Mexico City, include DF and some districts of the next states as Estado de México and Hidalgo. and all, as metropolitan zone have 25 millions aprox.
some NGOs about cyclist, organize in differents days critical mass, the most popular for the work to promove the cyclism, is with Bicitekas all wednesdays at 9pm in Angel de la Independencia monument, (Reforma Avenue).
and all the sundays, in the same avenue special to cyclists