I could go on forever about must visit spots in and around Moab. Its hard to go wrong there!
Go up to Island in the Sky...check out Green River Overlook, Grand Overlook. Take a stroll to Mesa Arch and look down on Washerwoman and Monster Tower. Drive down Shafer Trail if your into some 4wheeling (don't go down it if its just rained or going to rain). Visit Deadhorse Point State Park. Take a cruise down Potash Road for some easy to spot pictographs. Drive down Kane Creek to Hurrah Pass for a 4wheeling sampler. If you're not into driving yourself, get a guided tour. If you want a ride that will show you what Moab really has to offer, get a tour on Hells Revenge...or Behind the Rocks. DO NOT drive your tow vehicle on these. Modified trucks only. Want to ride a mountain bike? Check out the slickrock trail. Its unforgiving terrain so I suggest experienced riders only. An easy drive...that is incredibly scenic and is paved...go up Hwy 128. It follows the Colorado River and there are some spectacular views all along it. Check out the Fisher Towers...or better yet, the Priest and the Rectory that are south of it....along with Castleton Tower just east of it. Or take a drive up into the LaSals on a loop road.
On and on and on.
http://www.nps.gov/cany/index.htm
Many love Arches National Park. Its cool for sure, but I like Canyonlands Better. If you want relatively short hikes, Arches is the place.
Archview RV Resort a ways north of Moab could be ok. I've never camped there...just used their showers. I've actually never rv camped around Moab so defer to others for what is best in that regard.
There are numerous trails to hike in Zion but I've only done one as a descent off 1300' rock climb so can't help much. Get their park brochure for trip planning. Last I knew, you had to take shuttles into the park to ease congestion.
Another place you don't mention is Bryce. Its pretty cool too.
