Utah does not require off-road permits or stickers for road legal vehicles, like Hummers/Jeeps etc. (They do for ATV/UTVs.) No fees involved on 95 percent of the trails. Only exceptions are:
Some trails are accessed via Sand Flats, that is a county run area, and I believe it costs about 10 bucks for a week of access for the entire vehicle. This would be needed if running Hell's Revenge/Fins N Things/Porcupine Rim.
Entering a National Park such as Arches or Canyonlands. A seven day pass is about 10 bucks too. But very seldom does a group run in the Park. If you and your family decide to do Arches while you are here, you will have to pay the entrance fee.
All other travel is basically on Federal lands overseen by the BLM and no fees are required so long as the group remains under 25 vehicles. And that will never happen. Alrock, Neo, Dave, myself and most others know to keep the group smaller.
As for camping....a lot of people camp out at Neo's on a first come, first serve basis. If you want regular campsites, there is one right in town on the river, or the BLM has 500+ campsites in the area that costs between 10-15 bucks per night. No BLM sites have any hookups or running water; they do have vault or pit toilets. There are some commercial campgrounds in town that have showers, etc.
if the group decides to go into a National Park and as a group very seldom happens, or if you, want to take the family on a day trp to Arches (highly recommended, my opinion while you are here) andOnly time a permit is required is if a group of 25 or more vehicles are off-roading as a group and NEO, Alrock, Dave, myself, etc., will make sure that never happens. (Besides, that is too large of a group.)