Valley of the Gods in Southern Utah is a nice place to camp for the night
I'm going to second this. Valley of the Gods is like a miniature version of Monument Valley. It's managed by the BLM, so no entrance fees or BS to deal with. Camp up at Muley Point, above Valley of the Gods. It's a big bluff that stands about 1,000 feet above the valley floor, and you can camp right on the edge overlooking the San Juan River valley and Goosenecks state park. It's breathtaking, especially at sunset. The wind can get a bit gnarly up there and the overnight temperatures plummet in the winter (we camped there in February, and it dropped to 15 degrees overnight when it had been in the 60s during the day). Just watch a couple of YouTube videos about the place and know what to expect.
Some other things to see right in that immediate area would be Goosenecks State Park, and Mexican Hat Rock.
Monument Valley is also worth the visit, as are Antelope Canyon and Horseshoe Bend, if you find yourself near Page. Pro tip - do the "secret Antelope Canyon and Horseshoe Bend" tour instead of the regular canyon tours or the public viewpoint of Horseshoe bend. The "secret canyon" tour takes you to a different, smaller canyon, but it's just as breathtaking. It's on Navajo land and the tour company is only allowed to bring small groups - like 20 people at a time. Then when you're done in the canyon, they also take you to a private viewpoint overlooking Horseshoe bend, also on Navajo land. The other canyon trips and the main overlook at Horseshoe Bend can all get as crowded as Disneyland, especially during certain times of year.
Monument Valley is also worth the trip. The visitor's center is really cool, and it's just one of those things that you have to see in person in order to fully appreciate the beauty, like the Grand Canyon.