Here are reviews of the Moab RV parks. There are plenty of them right in town so you'd be close to your friends who are not camping. June will be hot so you'll want electric. There are many wonderful dry camping spots on BLM land but you would be more comfortable with electric.
http://www.rvparkreviews.com/regions/utah/moabThe only public campground with electric would be Dead Horse Point State Park at Canyonlands. It's very nice but a little more driving into town.
http://stateparks.utah.gov/parks/dead-horse/You're in for quite an experience with the Jeeping trails. They're awesome. We've done, among many other, Cliffhanger and Fins & Things with our little Jeep Liberty although we did have it raised and bigger tires. It did just fine. Definitely go with someone on any trail although the trails are well used and someone will eventually come by if you need help. I'd advise getting the Massey or Wells book on the area.
For hiking there are none better than the trails in Arches Nat'l Park. Try to get on the Fiery Furnace hike which is led by the rangers. You need reservations and it's a unique fun hike in the midst of slots.
http://www.nps.gov/arch/index.htmFor some cool temps, drive up to LaSal Mountain. The night Colorado River Cruise is fun, too. Your time will fly by. Have fun! It's a favorite place of ours. (We lived in Michigan at one time, too.)
http://www.discovermoab.com/