I'd recommend buying the Trailer Life Directory 20008 mapping software and bring a laptop. It will help you with route planning and show you where most of the parks are and how they are rated.
Personally, I'd plan the route for sites like the Grand Canyon, Bryce Canyon, Zion National Park and generally stay off the Interstate Highway system if not required (i.e. slower pace and more to see on the US and state highways).
My personal preference would be to go west from Zion through Las Vegas (stop if you must) and onto Death Valley which is very beautiful and not overly hot if you get through there before June.
From Death Valley I'd head over to US 395 and then go north to Reno.
From Reno I'd recommend heading up to Lassen National Monument and Mt. Lassen and take the back ways up to Klamath Falls, Oregon to intercept US 97.
From K-Falls I'd stay on US 97 all the way across the Columbia river to I-90/Ellensberg which is about 90 miles east of Seattle area. Note, US 97 doubles up as I-82 for part of the way in Washington.. If you have time I'd go further north on US 97 and cross the Cascades on US 2 which will bring you in on the north side of Seattle metro and all the way to Everett if you stay on to the end.
One thing to be aware of the is the bridge on US 97 where it crosses the Columbia River from Oregon into Washington is closed until May 23rd at 12:00 noon (current estimate). If you get to this area before May 23rd you will need to detour on US 197 to The Dalles and cross the Columbia there then backtrack on US 14 back to US 97. The detour is about a net 30 miles. You can go
here to check on the construction/closure status.