I would recommend you skip Oregon and Washington for this trip. You can always do them from California in a shorter trip of 2-3 weeks.
For this one, head to Phoenix for your must-do visit. Depending what part of the Phoenix area you want to visit, I'd suggest staying at any of the wonderful Maricopa County Parks. They have gorgeous sites, immaculate showers areas, electric, dump stations and at that time of year very few campers. You won't need reservations. Another great spot is Lost Dutchman State Park at the far eastern side of the Phoenix/Mesa/Apache Junction area. It's a beauty and no reservations needed in June. Then up to the Grand Canyon and stay at the Rim in the park at Trailer Village which has full hookups. It will be HOT in June and the rest of the summer.
Then go to Zion and Bryce National Parks and possibly Arches and definitely sign up in advance for the 'Fiery Furnace' hike for a morning. The kids will love it and you'll have some shade from the rocks. Arches and Zion will be hot but Bryce would be perfect. Arches does not have electric in their campground so plan to stay at a RV park in Moab. Zion has electric and also dry camping. Bryce doesn't have electric but it's cool enough that you won't need air conditioning. If you want hookups, Ruby's is just outside the park entrance to Bryce.
Then head up to Grand Teton Nat'l Park. With your size RV you could fit in any of the park's campgrounds. We like Gros Ventre but it is dry camping with a dump station and water fill. It, and Colter Bay campground each have 300 non-reservable sites and they rarely fill. Next to the Colter Bay campground is Colter Bay RV if you want hookups. A reservation is a must for the RV park.
From the Tetons you can head north through the park to Yellowstone. Fishing Bridge RV in the park is the only one with hookups. Madison is a very nice dry camping spot. It has a dump station and water fill.
I think by now your one-month will be up and time to head home.
