Arches NP is worth at least 2 or 3 days IMO. Others will know more about the Jeep trails around there.
June can be hot! If you become weary of heat, take a drive into Mani-LaSal NF.
A day trip into Colorado Nat'l Monument is a possibility, too.
It's been a while, but I think Portal is the place I camped at, too. It sits back from the highway enough so there isn't much noise. If you take your bikes, you can easily ride from this CG into town and along the numerous paved paths. There are bike riders all over the place, so drive attentively.