One of the reasons that people decide to move to another country is climate as well as the cost of living. Cost of living is really an important thing to consider when thinking about relocation.
Mexico is a big country and provides the ability to choose the lifestyle that suits you and your family. However, it's also very important to select your location within the country wisely.
If you stick to the big cities like Mexico city and a central location within a walking distance to all your interests then the price would be a premium. However, it will still cost much less compared to the USA or many other European countries.
Another thing to consider is how much time you are willing to spend to find housing. It's usually the best to find housing after you land in Mexico and do a local research.
However, today I would be talking about my experience and what budget I was living within while in Mexico City. Keep in mind that I was not doing a lot of research, I just booked apartments straight from Airbnb with a bit of negotiation.
Cost of living in central location of Mexico City
Apartment rent (1 bedroom): 700$ a month. Includes internet, water, and electricity. Cellphone: 15$ a month. "Paquette sin limite 300" from Telcel. Food: 400$ a month. Includes eating out every day at medium-priced restaurants. If you eat at taco-stands/street food it would be much cheaper. Insurance (optional): 40$ a month Other expenses: 100$ a month. Taking Uber, buying some clothes, unexpected spendings, etc.
Total: 1255$. This is the price that I spend by living by myself, without roommates and having my own kitchen and without signing any contracts with landlords.
Now let's take a look at how a price would change if we live in the same city by a much simpler standard and not in the central location.
Cost of living in non-central location of Mexico City
Apartment rent (1 bedroom): 300$ a month Cellphone: 15$ a month Food: 250$ a month. Includes eating at taco-stands/street food. Probably possible to stretch it even further by cooking yourself. Insurance (optional): 40$ a month Other expenses: 50$ a month. Let's say it's just an emergency fund.
Total: 640$. As you can see it's possible to have a budget almost twice cheaper.
What average budget I think that you should consider in Mexico city? I think that it's safe to calculate that 1000$ per month or a bit more would be enough for a comfortable life here for a single person. Not a luxury or anything else, but quite a comfortable life.
Of course, you could stretch your budget further or make a more comfortable living by just moving to a cheaper city within Mexico, a good example would be Queretaro.