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I would recommend the Polanco neighbourhood for your family. Its an upmarket area with lots of great shops and restaurants. Its also the main hotel area so you have more security around Polanco than anywhere else.
I have lived here for over 10 years and love it.
You don't need a car as everything is walkable. It's close to Gyms, supermarkets and several quality cinemas, museums and a brand new world class theatre.
You pay a little more for rent but for peace of mind and all of the above it's certainly worth it.
There are also 2 nice parks for the kids to play in as well as several good schools.

Hi Jen,

all depends, if you've never been in Mexico before there are different areas since Mexico City is a big city. It is indeed possible to live without a car and perhaps the best option if you live in a busy zone; the rush hour may extend for several hours.

I used to live around Napoles, it is nice there and you have options for your kids; other options are Tlalpan, you have a natural park with activities every weekend, you have two large shopping centres and good restaurants on Insurgentes Avenue. It is also an exit from the city, so you could easily go to nearby towns or even to "Ajusco", with outdoors activities for all the family, such as mountain bike, gotcha, fishing, camping, hiking, etc.

Coyoacan is also nice and full of schools, museums, cultural centres, restaurants, activities for kids, playgrounds.

Condesa is also ok but not many things for kids, there is only one park in the heart of the neighbourhood. San Miguel Chapultepec is nice and quiet, and Bosque de Chapultepec is very close with a lot of space and activities for your kids.

There are places like Naucalpan or Atizapan that are not part of Mexico City but Metropolitan Area; they have a lot of schools, shops, restaurants, parks and closed residential developments where kids can play safely. They have almost everything (no the charm of MxC, not many museums nor cultural centres nor colonial architecture), so basically you don't need to be in Mexico City, only if your job is based there.

I'd like to learn more.  Compare living expenses, please.