For day trips, here's a couple:
xochimilco canals - never been but it looks pretty neat. Kinda like a rural Venice, Mexican-style. To the south of town. I never went, but I might go there when I return to mexico.
Teotihuacan - a must on the tourist circuit - a short bus/taxi ride north of town. Massive pyramids that you can climb up.
Cuernavaca - mid-sized city to the south of DF maybe 1.5-2 hours away by bus/taxi, lots of expats, nightclubs and a nice little bohemian area downtown. Lots of spanish language schools here, too.
Tepoztlan/Tepozteco - maybe another hour outside of Cuernavaca to the east (I think), very strenuous hike up a mountainside and a small pyramid at the top. Supposedly a new-agey type hangout (which does not describe my style), but I enjoyed it (and ate some fried grasshoppers in the market). This would be a long day from DF by bus, maybe not so bad with a tour group or a taxi/rental car. I think the only public bus here is a 2nd class "chicken bus" - get ready for adventure if you take this route!
I've been to all these places except xochimilco. All worth a day trip, although you could easily fill up day trips with exploring DF itself (Chapultepec park/Anthropology museum, Zocalo/downtown, Palacio de Bellas Artes, a couple of the ritzy neighborhoods/colonias, bullfights, restaurants, etc).