Despite its historical significance, the route of US-66 is not the greatest if you're interested in seeing the spectacular scenery of the American West. The route of 66 specifically avoided the Rockies and Sierras, but those mountain ranges are now traversed by excellent roads. Depending on how much time you have, I'd suggest making a loop through the Colorado Rockies, Yellowstone, the Sierras and Grand Canyon. You could fly into Denver, and make that loop in around ten days without pushing too hard. Of course, if you want to take your time, you can stretch it out longer, or just skip a couple of destinations. You can easily spend a week in either Colorado or California without getting bored.
Rental RVs are certainly available, but not cheap and of course they burn a lot of gas. But a bigger issue arises if you leave the major highways. Driving a big RV over a typical 10 thousand foot mountain pass isn't much fun, what with narrow roads, steep dropoffs, hairpin turns and an underpowered vehicle. Honestly, you'll probably have more fun if you take a smaller vehicle and stay in motels, but that's up to you. Rental cars and vans are quite reasonable, as long as you opt out of the LDW (limited damage waiver) and other optional insurance. Check with your own insurance and your credit card company to see what coverage you already have.
Camping isn't much of an issue. There are a wide range of public and private campgrounds, the public ones are typically in state or national parks, or in national forests, and tend to be "rustic" (limited facilities). Private campgrounds are usually well equipped with amenities like showers and toilets, but you don't need that stuff if you're in an RV. A good thing to know is that most Walmarts allow RVs to park in their lots overnight, which gives you an out if you find yourself with no place to park late at night.
Another thing to consider when visiting the US is the time of year. Many mountain passes are closed until late May, even early June. June, July and August are the peak tourist months, prices will be high and popular destinations will be crowded. The worst weeks will be around the major holidays: Memorial Day (end of May), Independence Day (July 4th) and Labor Day (early September). After Labor Day, crowds drop quickly and mid-September is an ideal time to travel in most of the US.