You will love Mexico. Mexico City is fabulous, with so much to see and do. But after your 9 committed nights, do go out and see more of the country. Merida and San Cristobal are both lovely, but am wondering how easy/scenic it would be to "work your way back" to DF after flying to one of those locations?
Palenque is fabulous but time consuming to reach . fly to Tuxtla G and then on to San Cristobal and then to Palenque.
Have you considered Oaxaca? It's wonderful culturally - great architecture, museums, food. A 5-6 hour bus ride - but easy and direct and very comfortable on a first class bus. Very scenic most of the way as well. Lots of nature possibilities (small villages in the mountains/hikes/etc) around there too. You could also take 2 hour bus trip from DF to Puebla, explore it a few days and then go on to Oaxaca.

Another long bus ride - Morelia - a fabulous city also - and maybe take a day trip to see the monarch butterfly sanctuary?

Shorter bus trips - Guanajuato - a gorgeous colonial city and university town.

Agree with the Oaxaca rec. Food, ruins, music, museums..did I mention food? Puebla is nice as well, but not as nice as Oaxaca but worth a night or 2. Cholula has the big pyramid, but its mostly a mound of dirt with a church on top.
You could also go north, Queretaro, Tequisquiapan, Bernal, San Miguel, Guanajuato. Tequis/Bernal has has the wine and cheese route. In San Miguel you won't need much Spanish. Lots of English speakers in Oaxaca as well. If you end up in Oaxaca, you can pretty much see/do everything by taxi, but not so easy if you go north.
In Mexico City, Xochimilco is a hoot on Sundays.

Oaxaca is a good choice--easy to reach by bus.
Malinalco is a 'pueblo magico' just 2 hours SW Mexico City. It has a small Aztec ruin, lovely churches, a lively town plaza and market, and beautiful mountain scenery. You can get there buy bus from Terminal Observatorio. Go for 2 nights.

After a 5 day side trip last March to Guanajuato I have rented a house there for the Month of January. So I like it!

Hi:
I'm a veteran traveler to Mexico so here are some tips:
1. There's wonderful Saturday bazaar in Coyoacan.
2. The taxis are cheap so don't hesitate to get around the city with them. One reliable cab company is called "Sitio". Don't ever take the little green VW taxis.
Bus system to other towns is really good.
3. Definitely see the Centro, especially the Museo de Bellas Artes, the Museum of Popular Art and the National Museum on the Zocalo that has the Diego Rivera murals. The Zocalo is a happening.
4. The Antropological Museum at Chapultepec Park is wonderful too.
5. Nice neighborhoods for exploring are Polanco, Condesa and Roma. My favorite restaurant is Contramar in Roma and a reservation is usually advised. Expect to pay about $75 for two at most Mexico City restaurants.
6. Puebla, a gorgeous colonial town is about 60 miles south of D.F.and the bus goes there. Nice market there too.
7. YES to Oaxaca! It's gorgeous and the ride there on the bus.. 6 hours.. is through the spectacular Sierra Madre.Wonderful center with beautiful colonial buildings, lots of little shops and Monte Alban on the outskirts. You can also visit some places to see the black pottery typical of the area at roadside shops near Oaxaca.

Where are you staying in Coyoacan ? I've read that there are very few hotels, and then those aren't
listed ! We're going to the DF in April and would like to stay in this area. Thanks.

Puebla is a couple of hours by bus from Mexico, D.F. You might want to take a look at my trip report; click on my name to find it.

http://www.flickr.com/photos/mksfca/sets/72157623026204048/

Thank you everyone for your great input and recommendations!

Bedar, we are staying at a studio at a house right next door to the Frida Kahlo Museum which is nice and airy. I found it on www.airbnb.com.

After all your replies and some reading up, I am feeling very excited for our trip!

We are currently living in DF and LOVE it. Some additional thoughts not already hit upon:

San Angel has a great Saturday bazaar (lots of local artists and great finds). It is right next to Cayoacan. You could easily get a taxi there from Cayoacan. I highly recommend San Miguel - there are "luxury buses" that run from Mexico City up there (A/C, TVs, bathroom) - I haven't used any lines myself, but you can probably find websites easily online (you will want executive class). Taxco is also a quaint city - lots of silver, cobblestone hills, etc. You can do a day trip from Mexico City easily. We haven't been to Oaxaca yet, but it is definitely on our todo list before we go. You will also be here during the migration of the Monarch Butterflies - so if you are interested in seeing them (which is very cool in my opinion), that is another easy day trip from the City. The place I've gone is called Piedra Herrada Sanctuary - close to Valle de Bravo. If you want to have some easily organized tours - look into the major hotels in the city - they all offer group tours to the top destinations in the areas. My parents did this through the Marriott when they were here and were happy with it.

In terms of tips on safety in the city - I have found it to be quite safe (but I'm a city dweller, so am used to large cities). I also live in Polanco, which is a unique/safe area. Just be smart. Try not to look too much like a tourist. Don't carry things in your purse that you wouldn't want to lose (i.e., passport etc). I have a small wallet that I clip to my pants where ever I go so that if my purse were snatched I wouldn't be devastated. Another tip is to carry some cash in a pocket as "throw down cash" in case someone tries to mug you. Now, that said, neither I nor any of my friends residing here has ever been mugged. I echo the post regarding taxis - ONLY take taxis from Sitio stands - there are stands in most neighborhoods around the city. Always ask the fair before you head out.

Best of luck and enjoy!