Mexico City

I stayed in Condesa (as a solo female traveler) and felt very safe. Lots of restaurants and cafes, nice parks, people out and about. The other neighborhoods that are mentioned frequently as being safe/good for visitors are Polanco (lots of upscale hotels and restaurants in this area - seemed nice to me but I preferred the feel of Condesa, would probably be very safe, though), and Roma Norte (seemed hip, for lack of a better word, lots of bars and restaurants, maybe a little gritty).

I also really liked Coyoacan, the area near the Frida Kahlo Museum. No idea if it’s considered safe or not, but it felt safe to me - lots of cute restaurants and cafes.

Here's a link to a current thread that may help. https://community.ricksteves.com/travel-forum/trip-reports/a-week-in-mexico-city

The thread Allan linked to has a lot of good info. The hotel I've stayed at is the Gillow, in the Centro Historico, between the Zocalo and the Alameda. It's a busy and safe neighborhood, close to some of the major sights, including the Aztec Templo Mayor. The Metro is a good way to get around beyond walking distance, so are taxis at an official stand (sitio) or called by your hotel or restaurant. Condesa and Roma, a little farther from the center, are also good neighborhoods to stay in.

The major "safety" issue is pickpocketing, which is an issue in most big Latin American or European cities. Our host's advice applies to Mexico City as it does to Europe. Using a money belt or neck wallet, and being aware of your surroundings, are the best ways to avoid this problem.

I went to Teotihuacan with a tour company that I wouldn't recommend because they ran us through too many "shopping opportunities" on the way there. But I don't remember the company name, so that isn't much help. That said, they got us there and back safely and on time. You might look to a guidebook for advice here, or posters yet to come.

Back in 2018, I found that taking Uber all the way to Teotihuacan was very reasonable. $30 or so each way, without having to deal with the metro+bus station. I was ready to take the bus back, but surprised to find that Ubers were also available at Teotihuacan without issue.
It did cost more than the bus of course, but between 2 people and compared to US or Western Europe costs, it was a bargain.

Thanks everyone. You have been most helpful.

In September 2021 we visited Mexico City for several days. In the historical center the Hotel Ritz was enjoyable. We then moved to the Hotal Laila in the more modern part of downtown. We were happy with both and felt safe during out trip - including the area around the Freda Kahlo Museum. For Teotihuacan try: Great Tour to Teotihuacan and Shrine of Our Lady of Guadalupe at www.wayak.mx. It was a small bus tour and guide Franando Guzman did a great job.

We stayed in the Condesa in an Airbnb for 6 nights in 2019 and it was great. We felt safe in the evenings with local families out walking.

We went to Teotihuacan on the public bus but you could take one of the tours that leaves from the Centro Historico hotels or Uber as suggested by another poster. Teotihuacan is amazing.