Mexico City is one of the largest metropolitan areas of the world. More than twenty million people live here, from some of the world's richest to uneducated peasants just in from their busted farm scratching out a living at the rubbish dumps sorting trash, parts of the Estado / Edo. de México (state of Mexico) with significant crime profiles.
World class museums, parks, hotels and restaurants, entertainment, archaeological sites, palaces, lakes, the oldest zoo in the Americas (predating the arrival of European conquerors), easy access to resorts, wonderful archaeological sites, colonial architecture and varied cultural experiences by air or deluxe busses with fully reclining seats and wifi.
Like any big city, Mexico City has dangers and annoyances for visitors and for those living there. Crowds, poverty and relative wealth mean pickpockets and bag snatchers of various kinds where visitors in particular tend to congregate. That's true here, Barcelona, Paris, etc.
I've been a denizen, and still have friends and family who are "Chilangos" (people who live in Mexico City). Just saw some of my "Chilangería" last week in "the safest Mexican city", San Antonio, Texas
, so-called because of the substantial number of Mexicans with five year visitor cards and purchased homes there. Many of them are from Monterey, the industrial northern city in the state of Nuevo León, which has experienced some personal safety problems of late.
In Mexico City, pirate cabs are a possible danger - "express" kidnappings (they use a weapon and threat of violence to make you draw cash from a cash machine / ATM and end up dumping you in an unsavory area of town minus watch, wallet, luggage) are often perpetrated by pirate taxi operators and their henchmen. Use a taxi from a "sitio" or taxi rank; restaurants, shops, friends etc. will always be happy to call their nearby sitio to get you a safe taxi. Don't take roving "taxis" on the street, e.g. outside the airport (unless you're sure they are taxis and you know what you're doing. ("Peseros", shared taxi vans with assigned routes and therefore cost much less, are probably safe - but they can be pretty crowded, and some drivers are, shall we say, "improvisational" in their approach to driving.)
Other public transit can be good - the express trolleybusses are fine - but under crowded conditions can be uncomfortable; the metro at its most crowded times has opportunistic theft and groping reports with disturbing regularity.
I'll add more and you might have some experiences to share.