My family of 4 (and dog) are planning a RV trip starting from Las Vegas. We plan to stop at the Grand Canyon, Zion National Park, Bryce Canyon National Park, Monument Valley and Lake Powell. We would love some suggestions for good campsites. Thanks!
Monument Valley Navajo Tribal Park doesn't offer many options. There is Goulding's Lodge: Campground, located close to Goulding Lodge. Showers are not open 24 hrs. I heard about a new campground being built, but I don't know if it's operational yet, or if it is tent only.
If you have kids with you, Zion Ponderosa Ranch Resort could be an option. It is a little bit of a drive out the east entrance of Zion National Park. Zion Canyon Campground in Springdale is also an option and very close to the action. The camping inside the park at Watchman Campground is in an amazing location, but very popular and fills quickly. People book months in advance. If you are traveling between April and the end of October, you will not be able to drive your own vehicle into the Zion Canyon portion of the park. You will need to use the free bus shuttle service: http://www.nps.gov/zion/planyourvisit/shuttle-system.htm
There are not many options at the Grand Canyon South Rim. There is Mather Campground inside the park at Grand Canyon Village and Grand Canyon Camper Village in Tusayan. Mather is the nicer option and more popular. Both will often be at 100% occupancy, so book ahead.
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If you're into free camping and can make do without a bathroom/showers, there is free BLM land that is popular to camp in about 20 miles outside of the Zion park (doesn't seem that far). You can drive south from the south Zion entrance and at Virgin there's a Kolob reservoir road that goes way up to a reservoir (and from my visit one of my favorite parts of the park that I visited was driving out this road). Take that road about 5 miles and there'll be on the right along the river signs for "Camp by Donation" or you'll just see people camping with their vehicles and you can drive down and find a place for free! More private than in a campground and quite nice. The roads are a bit rough so having a little clearance under your car is helpful otherwise be careful which roads you navigate. Hope that helps! Free camping just outside of Zion is hard to beat!