You may want to stay in the traditional campgrounds because of the kids, but if you want a more private experience.......
Find the USFS District Office for the area you are in, they can point you to the dispersed sites in the National Forests. Often, the OHRV maps show the areas for these sites. They sites are quiet, and free and located in beautiful but much less crowded areas. Don't forget your shovel though....
here's some Grand Canyon Pics on the Kaibab trail from south rim to the Colorado River from Jan 2005....
Half way from the south rim to the COlorado River
Finally, the river is visible and the last steep section.
River is getting closer.
swmbo and i had our honeymoon in Moab.
We did arches and hiked the sliprock to delicate arch and also went to dead horse point*.
We alo hiked to various petroglyphs and pictographs site in canyonlands.
This was in November so we kind of had all the sites to ourselves.
Lots of beautiful sites. I still have some good books like the falcon guide to Arches and Canyonlands. If you want me to send them top you,let me know.
You need to read or reread Desert Solitaire by Edward Abbey, America's beloved anarchist.
In the early 70s, I camped out of a VW bug doing a cross country perambulation and ended up in NW Utah and in the middle of nowhere and found a great campsite in a red rock canyon. Perry is correct thatsome of the NFS sites are real gems.
Since you habe been a flatlander,make sure your breaks and tires are up to snuff for your safety.
* Dead horse Point is where they filmed Thelma and Louise riding off into the sunset. 😐
Kelly and I at the south rim, 1992
( I have a habit of squatting down for pics to make me look normal) doesn't work!