The route we would follow is I-10W to I-49N in Louisiana. Then take that to I-20W and follow that to Dallas. Get on US-287 to Amarillo. Just south of Amarillo is Palo Duro State Park which is a good place to spend a day or two. They have a good dinner play theater that is worth going to. Then back up to I-40W. If you like urban areas then check out Albuquerque, past that there is Chaco Canyon to the north, you don't say what RV you have but that road into that park is rough dirt.
There are quite a few state parks on both sides of I-40 and El Malpais NM if you want to see a lot of lava. When you get into AZ then you'll hit Petrified Forest and further along Barringer meteor crater.
You'll pass Walnut Canyon (Indian Ruins) on the way to US-64 to Grand Canyon South Rim. Follow 64 out of GCNP to US-89 follow that up into Utah passing some other small National Parks/Monuments along the way. Once in Utah, check out Coral Pink Sand Dunes SP and then Zion->Bryce Canyon-> Arches/Canyonlands -> Capital Reef. You can divert east into CO for Mesa Verde the best of the Indian Ruins type parks. Hovenweep NM, Goosenecks SP, Natural Bridges NM, Valley of the Gods and Monument Valley are all in the same area and Cortez CO is a good central point to camp while seeing all of them. Monticello UT is a good spot to camp for the Needles District of Canyonlands.
We've done this any number of times starting from north Florida. 300 Miles/day is a good target. Some of these parks (Bryce in particular) are at high altitude and snow is possible in November.
