I have a part-time place in Green River and let me tell you, it's HOT, as in over 100F. If you come down here, like Gary says, be prepared. The Green River State Park is one of the nicer places in the summer because it's next to the river and has lots of big cottonwoods, but it will still be unbearably hot. And you would think all the tourists would be gone, but that's not the case, as the Europeans come in the summer. If you want to be cool, go up in the La Sal mtns, but even up there it gets hot. If you have a rig and can get hookups, that's the way to go. I used to be on Grand County Search and Rescue, and believe me, hiking in the heat is a quick way to meet your maker, drink lots of water.
As for going into the Swell and places like the Wedge, be prepared this time of year to take care of yourself, as there are lots fewer people there than in Moab, and the dirt-bikers who frequent the area are usually elsewhere than in the heat. I've been up there many times in the summer and seen almost no one. Lots of places with no cell coverage.
But...it's a GREAT time to go rafting. The rivers are running high. And I will add that summer in the desert country around Green River and Moab is my favorite time to camp, as the nights are to die for, they're so balmy. But you have to make it through the day first. And be forewarned that the gnats are out and really bad this year.