Remember that some of these parks are very hot in July. Moab especially can really bake. Zion is also quite warm. Plan your hiking early and late in the day and find someplace cool midday. Do get reservations early, as the week of July 4th is always packed.
You can do the following loop in either direction.
Take I-70 across CO to Moab. Spend several days there. Arches may be a PITA with the construction next year, but head in at least as far as the Windows section. If you have the time, consider taking the Fiery Furnace guided walk. Real fun for kids. Take an evening hike up to Delicate Arch to watch the sunset. Take a couple of headlamps for the trip back down to the car. For Canyonlands, head up early and drive Island in the Sky, doing the short walks out to the various viewpoints. For your trip back to town, drop down Shafer Trail near the entrance and go through Potash and along the river. Some interesting petroglyphs along there (stop at the second turnout for the better ones). This will probably take close to two hours from the top of Shafer, so plan accordingly.
For midday adventures, try hiking Negro Bill Canyon (a few miles up UT 128) to Morning Glory Arch. A narrow canyon with high walls for shade and a small creek to dip your feet in to keep cool. Mill Creek above Moab also has pools to swim in. Or take a raft trip on the Colorado. At some point, driving up (or down) UT 128 for 10 or 15 miles along the Colorado River is a gorgeous excursion. Or drive the LaSal Mountain Loop. You'll get a great view of the red rock country from up high. Nice views and much cooler temps up there.
The return to I-70 to UT 24 etc. to get to Capitol Reef. Some driving in this park, but it will be hot in mid summer for some of the hikes. If you can do without electricity, you might consider camping at Singletree NFS CG. It's up high at 8200' with tall pines, so much cooler than down in the desert. From that CG, you can not only explore Capitol Reef, but also part of Grand Staircase Escalante NM. Do a search on Spooky and Peekaboo slot canyons and watch some videos. Great fun for kids. The hike up to Calf Creek Falls is a great midday thing to do. Swim or wade in the pool below the falls. But don't turn in here with your trailer as it's not at all friendly to towed vehicles.
From there, head to Bryce. We stay at the CG's in the park. Again, at 8000', Bryce is higher up so cooler with lots of shade from trees. Use the shuttle and drop in one trailhead and come out another. Mid to late afternoon hiking isn't bad here, as the plateau shades the trails.
Then it's off to Zion. With kids, I strongly recommend camping in the park at Watchman CG. They do have electricity and a dump station. The Virgin "River" flows by the CG. In July, it will be a slow flow. Kids get in there and build dams to create ponds of water to splash in. Great cottonwood trees overhead provide shade. Take your folding chairs and sit out in the river. A great way to keep cool during the heat of the day. Lots of hiking in Zion. Plan your hikes by doing the east side trails in the morning and the west side trails in the afternoon so you can stay in the shade. A hike up Zion Narrows will be much cooler, even midday. Bring some water shoes or tennis shoes you don't mind getting wet as you'll be wading up the river for much of the hike. A great excursion for the whole family.
Then I'd head to the South Rim of the Grand Canyon. Stay in the park, use the shuttle. In mid-July, I wouldn't plan any big hikes into the canyon. Maybe drop down the Bright Angel Trail a few miles. If you go in the afternoon, the upper part of the trail is in shade. And remember, once you get back to the top, the ice cream stand is right there to replenish those calories you used up.
Now personally I'd skip Petrified Forest. I'm not impressed (and I'm a geologist!). Also, it's a bit out of the way on this loop, especially given your limited amount of time for this trip. Instead, I'd head up US 160 to Mesa Verde. Camp up in the park which is higher in elevation. There are a limited number of sites with hookups there, but reserve early. Get in on one or two of the guided hikes into the cliff dwellings. But make sure you also do the Mesa Top Loop and see some of the surface ruins.
Then off to Durango. A must-do in my book is the Durango to Silverton Narrow Gauge Railroad. Takes you through some incredible scenery. At one point, you'll have wilderness on both sides. And Silverton is a real honest to goodness old mining town.
You could do a raft trip out of Durango or tube the river flowing through Pagosa Springs.
From there, continue on US 160 over Wolf Creek Pass. If you have time and desire, it's not a terribly long distance off the highway to Great Sand Dunes NP, although the sand can get pretty hot midday in the summer time.
If you do the loop the way I've laid it out, you'll get to the cooler temps of the mountains after you've suffered through the heat. If you go the other way, you'll have the coolness first and might find the heat a little more difficult to handle.