RV camping can be fun almost anywhere, but I think it's best in the west. You can drive into the desert in many places and stay there for free for up to two weeks. Most Bureau of Land Management (BLM) lands, national forest lands, and many state forest lands are affected by this. Every two weeks, you have to move, but how far is up to you, and most of the time, your choice will be accepted.
Long-term camping in RVs on BLM land
"Boondocking," which is camping in an RV, is becoming more popular. In fact, the BLM has started to set up special areas where people can stay longer, especially in Arizona. A permit costs about $140, but you can stay for up to six months and get water, pump stations, and dumpsters. In some of these places, people live. It costs less than property taxes or rent for a parking lot.
Arizona has a lot of RV camping in the winter. Quartzite is one of the places where the most "boondockers" get together. At least part of the year, several hundred thousand people live here in their RVs. It's on Interstate 10 close to the California border and only 20 miles from the Colorado River. Quartzite is known for its gem shows, swap meets, and the fact that its population grows every winter. It is surrounded by BLM land.
If you ask around in the desert southwest, you'll find out that every winter, RV communities form. Some of these temporary towns, like "Slab City" in California, are run by RVers who run businesses like bookstores, grocery stores, and other shops. When summer comes, these "boondock" communities go away, but they come back the next winter.
Other places you can camp with an RV
If you look around, you can find "hidden" places in the desert southwest where you can park your RV for a week or a month. Some are cheap, and others are free. For example, the Hot Well Dunes Recreation Area north of Bowie, Arizona, costs $3 per night and has nice hot springs and lots of wildlife. A permit for the whole year costs $30, but you can only stay for two weeks per month (permits are sold at the BLM office in Safford). You can stay for free outside the fence, but then you can't use the hot springs or picnic tables with shade.
Contact the Bureau of Land Management to find out about other places. They can tell you what you can do that falls under their control. Also, Woodall's Campground Guide has a list of free campgrounds. Keep an eye out for other RVs that are parked in the desert or forest. Lastly, talk to people. Most of what you need to know about RV camping will come from other RV campers.