Here is a listing with descriptions of many of the trails. Don't know what trail rating you can handle? Check out the ratings
here.
Can't help you with specific recommendations on CG's as we don't use hookups, but if you do a search here (the box above), you'll find well over 150 posts in the last year. This is one of the most requested places for advice on this forum.
Do remember in June it has the possibility of being pretty hot. Make sure you (and your jeep) are both ready for 100+ degree days. There's so much to do around Moab. We spend three to four weeks a year here and still have things on our list we haven't done. Lots of great hiking in Arches, rafting, biking, etc.