I always recommend Navajo National Monument (Arizona), which celebrates the Ancestral Puebloan culture (previously Anasazi), but is in the middle of the Navajo reservation. Betatakin Ruin is the most beautiful and photogenic of all the cliff dwellings, IMO; the setting is just stunning.
No guided tours to the cliff dwellings between Oct - April, but there are things you can do on your own. To me it is a special and beautiful place. I used to use it as a half-way camping spot between Dolores, CO and Flagstaff, AZ when going to visit my ex, who was in grad school in Flagstaff in the summers.
They have two camping areas, one somewhat more developed than the other. Beautiful setting, with red rocks more typical of Moab (Utah) and Gallup (NM) area.
http://www.nps.gov/nava/index.htmHowever, it is relatively high elevation (~7,000') so check the weather first.
Photo gallery (not mine) of Navajo National Monument
here. Yes, it IS that amazing.
Villanueva State Park is also one of my favorites . It is a modest little park in a small canyon right on the Pecos River. There is a hike up on top of the mesa to an archaeological site. Again, this is New Mexico (relatively high elevation), so check the weather. The winding road along the Pecos River that connects the two interstates is like a trip through time, New Mexico style.
If going the southern route, never miss Carlsbad Caverns!