Your trip will be too late in the fall to stay in any Forest Service campgrounds in Colorado--they will be closed. You can stay at Colorado state park campgrounds, however, which have 30/50A electricity (so heat and light for the animals) and are pet-friendly. But if you don't have time to sightsee, there may not be much point in visiting Colorado.
If you do have a little time to see sights, I would suggest taking I-70 to Limon, Colorado, then US 24 to Colorado Springs. Stay at Cheyenne Mountain State Park in Colorado Springs, then take CO 115 to Canon City. You will join US 50 there and can take it through the Arkansas River canyon (Royal Gorge) and over Monarch Pass to Black Canyon of the Gunnison National Park. Ridgway State Park between Montrose and Ouray has very nice electric-only or full-hookup sites for $22 to $24/night. Then travel CO 62 and CO 145 to Telluride, Dolores, and Cortez, with a stop at Mesa Verde NP. From there you can travel US 160 to Tuba City, AZ, and the Grand Canyon, then I-40 to Hoover Dam and possibly Las Vegas, as suggested previously.
You may run into a little snow on Monarch and Lizard Head Passes, Cerro Summit, and Dallas Divide. But that time of the year, you should be able to drive any of them without problems the day after the snow falls because road crews have to keep these routes open all year. And you may get some spectacular photos of the Collegiate Range and the San Juans if they have a fresh coat of snow and the bright sun and blue skies that usually accompany clearing weather.
If you can't wait out a storm, head south from wherever you are (or west from Ridgway toward Moab, UT) and you should avoid any problems. But I would only visit Colorado on your way west, not on the return trip in November or you may see more snow on the passes than you really want to see.
On your return trip in November, I-40 around Flagstaff could have snow due to its high elevation (7,000 ft.). You might try I-10 through Phoenix and Tucson, as that is about the time of the year that many "snowbird" RVers arrive there. If you go that route, we suggest that you try one of our favorites--the Butterfield RV Park in Benson, AZ--and sign up for their nighttime telescope viewing session.
Depending on the weather, from Las Cruces, NM you could head north on I-25 to I-40 at Albuquerque or stay on I-10 through El Paso. In Texas, you can work your way north from I-10 onto I-20 to I-30 to I-40 in Arkansas and finally to I-81 in Tennessee. That will keep you south as long as possible along a relatively straight route. There are also a number of well-maintained Texas state parks that have electrical hookups along those Interstates at Monahans, Big Spring, Colorado City, south of Abilene, Mineral Wells, Mt. Pleasant, and south of Texarkana. In Arkansas, Lake Catherine State Park is 12 miles north of I-30 near Hot Springs and is a beautiful, forested, classic CCC-built campground with full hookups for as little as $19/night (with 30A electricity).
However, no matter which way you go, you will see some great scenery. So, have fun planning your trip and traveling in your new RV!
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2015 RAM 2500 4x4 crewcab 6.7L CTD; 2016 Rockwood Signature UltraLite 5th wheel