We love the North Rim of the Grand Canyon because of the quietness and more of an intimate feeling but if this is your first visit, go to the South Rim and stay at Trailer Village, F/H/U right in the park near the rim's edge and the free shuttle bus stops at the c.g. entrance - an excellent way to see the park.
Don't try to do Zion and Bryce from one campground - too much driving.
Zion - stay in the park at Watchman c.g. Electric and dump and water fill. Easy access for the free shuttle bus and visitor center.
Bryce - in the park - dry campground or for F/H/U - Ruby's right outside the park entrance. We love Red Canyon forest service campground and some of the sites have electric. It's about 5 miles from the park entrance.
Have a good trip!