Mexico City

Anyone associated with the visitor industry will likely speak English. The population at large...well, maybe not so much. As is the case everywhere, however, if you learn a few basic phrases then you'll be fine.
Trip Advisor has a pretty good "things to do" menu that would probably be of help to you, both to see what your options are (thousands, literally) and help you sort things out according to your particular interests.
We really enjoyed the National Museum of Anthropology in Chapultepec Park. Only scheduled a couple of hours there and could have spent a couple of days exploring the place - it really is terrific. The park itself is great too. Has some very nice wooded walks. Lots of history too, especially if you read ahead a little about the Mexican War and familiarize yourself with the geography of the woods and the actions that took place there. Some of the monuments memorializing the Mexican defenders of the city are quite moving.
A tour out to Teotihuacan is great, but can be quite hot depending upon the season.
Believe they also have day trips out to the Butterfly Sanctuary up in the mountains, but you'd need to research whether the little guys would still be in residence during your visit. They winter-over before taking wing and starting their reverse migration...in May I think.
We also enjoyed strolling up Paseo de la Reforma - beautiful, tree lined street of monuments and lots of little artisan shops along the way.
And the subway is cheap and efficient (and safe) for getting around.

Been to Mexico City a few times, it's a very underrated city. Outside of hotels, there isn't much English spoken:(
But do your research via guide books and web resources and you will be fine. But if you find yourself overwhelmed, you can easily have your hotel book day tours.
Must sees includes the amazing pyramid complex outside the city; it was more amazed by them than the ones in Cairo. The Guadalupe Basilica; outside of the Vatican, no other Catholic site receives more pilgrims. Two other must-sees includes the Templo Mayor, and the Anthropology museum. Something to also consider is a soccer match at Azteca Stadium. It's one of the largest stadiums in the world and the atmosphere is one of the most electric. Tickets can be easily purchased via Ticketmaster Mexico.

The city metro system is one of the largest, and cheapest in the world but can also be very hot (no AC) and crowded during rush hour. Consider using taxis during busy hours. Also beware of pickpockets and purse/bag slashers on the subway. The crowded conditions make ideal.

The locals are very friendly, some of the kindest I have ever met in all my travels. Those that speak english are very eager to help out.

I highly recommend the pyramids of Teotihuacan outside of the city. You can see the ancient site on a day trip.
https://en.wikipedia.org/wiki/Teotihuacan

Also, the Ballet Folklorico is excellent.

There will be plenty of people that speak English, no problem there.

Make sure that you don't take a taxi from the street, have the hotel call you a taxi.

Mexico City is relatively safe in the main city areas, but it is wise to be careful.

Caveat: I have not been to Mexico City but have done tons of research on it because I would like to visit in the next year or so. There are so many sites to see that I would be hard pressed to try to fill them in 7 days (even 12 days), especially given how spread out the city is. I'm a major "museum person", so I could spend an entire day (or several) in the Archeological Museum...but there are many, many other museums in Mexico City that are lesser known than the world famous Archeological Museum (obviously the Frida Kahlo Museum/Blue House is a major highlight too, among many others). I would not expect people outside of major hotels to speak English. I've been to other non-touristy (by Americans at least) areas in Mexico and it was immensely helpful to know basic Spanish phrases...it made my life so much easier when dealing with transport, places to eat, and talking with locals, etc. I would not assume that it will be like Europe. Hopefully that won't keep you from visiting local areas and neighborhoods and trying things on the menu that are unfamiliar (crispy grasshoppers, anyone?). I would say visiting the local food (and other) markets are a "must". If it were me, I would definitely try local places (outside the tourist zone and hotels) to eat, even street food (with some precautions, of course). If you were staying for longer than 7 days, there are also several worthwhile side trips from Mexico City but I would leave them for another visit.

This is an excellent book on Mexico City..it is from 2003 but extraordinarily comprehensive. I don't think it will make your choices any easier though :-)
https://www.amazon.com/Mexico-City-Complete-Tourist-Guide/dp/9681912985/ref=pd_sim_14_30?_encoding=UTF8&pd_rd_i=9681912985&pd_rd_r=9Y4HA3H9887H219HTQZD&pd_rd_w=SpNTa&pd_rd_wg=hWLqd&psc=1&refRID=9Y4HA3H9887H219HTQZD

This is also a good book with lots of recommendations and walking routes:
https://www.amazon.com/Mexico-CIty-Opinionated-Curious-Traveler/dp/1511448334/ref=sr_1_1?ie=UTF8&qid=1491346554&sr=8-1&keywords=Mexico+City

Thank you all for your suggestions! This is so helpful, I am printing your responses so I can refer to them when planning a framework for an itinerary.

Thank you all for your suggestions! This is so helpful, I am printing your responses so I can refer to them when planning a framework for an itinerary.

I was in Mexico City a few months ago and used Uber for all my transportation, it was inexpensive and easy. I took 8 different rides with no issues but don't under estimate the amount of time it will take you to go even short distances, I have never experienced such congestion. I would also recommend visiting Xochimilco and renting a trajineras, it's very crowded on the weekends with locals but I think that made the experience even better. Frida Kahlo's house is also interesting but get your tickets ahead of time, it always seems to be busy.

I hope you will do a trip report once you're back - I'd be very interested to know how things went. I look forward to going there myself in the next few years.

Hadn't thought about using Uber, that's a great idea. I was hoping we could walk to a lot of places but have no idea how pedestrian friendly the city is. And to Agnes, I will definitely post something about our adventure!

I'm sure there are many distinct pedestrian areas/neighborhoods, but the issue is the entire city is very, very spread out...so depending on where you want to go, you should count on using some combo other than by foot alone. The same is true in Manhattan (and Mexico City is a bigger, more spread out version of it). That's why both cities have an extensive subway/bus system. When I go, I'll seriously consider using two hotel bases so I can cover more areas on foot/short subway trips.

Your hotel is right on the Zocalo in the heart of the historic center, literally in sight of the National Palace (Diego Rivera murals), Templo Mayor, and Cathedral, a short walk from other sights like the Bellas Artes, House of Tiles, Torre Latinoamericano, and the Alameda. A short subway ride will take you to Chapultepec Park and the world-class National Anthropological Museum where you could spend days, as others have described. Teotihuacan and the Guadeloupe shrine are north of the city and perhaps best seen on a van tour, which your hotel can set up. The red hop-on-hop-off bus tour is a good way to spend a day, and will take you to most of the sights in the central city plus the Frida Kahlo house and the Trotsky house in nearby Coyoacan.

It's a big, busy, crowded city, full of life and variety, too spread out to be entirely walkable, but with good public transportation (best avoided in rush hour) to places too far to walk, and tours to places still farther out. You'll have no trouble filling a week there.

Thanks for sharing your thoughts and suggestions, Dick. No one I know has been to Mexico City so this forum has been immensely helpful. We leave Thursday and can't wait!