Can think of many places to go in June/ July, other than Utah, unless you are headed to Northern Utah. Salt Lake City/ Park City has much better weather that time of year.
In our 15 years of Utah experience, the best time to go to Bryce, Zion, and Arches is September and October. Leaves are changing, gorgeous weather, and no crowds. June thru August is pretty darn hot.
Have a great trip. We are a couple months ahead of you, and are headed for Oregon, Washington, and Montana for the Spring/ Summer this year.
Safe travels
