We've done that trip many times. Usually we head down I-10. In my opinion there isn't a lot to see until you are past San Antonio. There are some interesting Texas state parks tho. Palo Duro is one you should see. Carlsbad Caverns is the first major park on the trip, White Sands is in the same area. Traveling west brings you near Chiricahua NM in AZ. A little visited park that is worth a few days. Bisbee and Tombstone are in that area and are worth a day trip.
North on US191 brings you by Petrified Forest and Canyon De Chelley. Cortez CO is a good base camp area for many sites: Mesa Verde, Hovenweep, Monument Valley, Valley of the Gods and lots of smaller Indian ruins type sites.
This puts you in the Four Corners area which is packed with major parks: Bryce, Zion, North Rim of the Grand Canyon, Canyonlands and Capital Reef. I don't know about spending a month at one of them but we have spent 7-10 days around all of them at one time or the other. There are many smaller sites, too many to list but some are Kodachrome SP, Natural Bridges, Petroglyph SP, Pink Coral Sand Dunes SP and so on.
I'd plan on at least 1 weeks stays at Cortez, Moab, Torrey, Bryce, Zion and North Rim if you have the time. This assumes you get out of your vehicle and explore the areas. If you like slot canyons then a week at Escalante UT is worth it.
