UHH map by David Casteel, on Flickr

I can provide detailed driving instructions, but they are much too big to put here.

Unless I had a good reason for that order, I'd probably go first to Portland, then to Glacier NP, then to Arches NP (Moab), and then to the North Rim. And I'd take the coastal route to Bakersfield, CA and up to Portland. That route does not travel the same roads--it's about 300 miles longer, but does offer some more possibly interesting sites.
UHH map2 by David Casteel, on Flickr

And as long as passing that area, I'd probably add Yellowstone and Grand Teton NPs, too.
UHH map3 by David Casteel, on Flickr

These maps were developed using MS Streets & Trips, a tool no longer available, unfortunately. I like it and will continue to use it as long as it will work on my Win7Pro computer.

Columbia River GorgeYou do not say what month you are traveling. There area a series of three waterfalls near Multomah Falls; all situated along the Historic Columbia River Highway which parallels i-84/US-30 right next to the Columbia River.

From West to East the are Bridal Veil Falls, Multomah Falls, and Horsetail Falls. Multomah Falls has a parking lot accessible directly from I-84 however you cannot get (to my knowledge) from this parking lot to the old highway.
The old highway will not accommodate a wide RV or a long trailer and it gets VERY narrow in spots.

Also during the summer months, Multomah Falls is jam packed ( we visited in March 2022) and reservations may be needed. The other two falls had far less traffic.

The Historic Columbia River Highway is a 7-8 mile long stretch with entry and exit points to I-84 at both ends. Multomah Falls is located ~30 miles East of Portland.

Other sites:
Dinosaur National Monument: This has a wonderful little office by the parking lot what has a stupendous and little known follis display. Sometimes a range will be working on newly found fossils in the central room.

Yellowstone Nat'l Park: A gem with something for everyone. My favorite location is Mammoth Hot Springs and the
Travertine Falls. Light is best just before sunrise or else in the latter part of the afternoon and evening when the sun is not directly hitting the white and variegated calcium carbonate. This is a living area and it changes every year as springs dry up and new springs open up. Color is from mineral deposition and some mosses. Of course Yellowstone can be busy but the hot springs are far from Old Faithful. Gardiner, MT is nearly with many eateries and motel rooms. Livingston, MT is 50 miles to the North and right on I-80. A newish Fairfield Inn & Suites is located there and there are a couple of good steak houses (Montana's Ribs & Chop House).

The Palouse: This agricultural area glows in the spring. I think the favorite vantage point is StepToe Butte south of Spokane.

Upper North Falls @ Silver Falls State Park: waterfalls with not too long of a hike. ~ 30-40 miles ease of Salem, OR.

Glacier Nat'l Park: Going-to-the-Sun RoadP now requires a timed entry ticket. IO think the tickets are good for three days. More info on the link.

Oregon Coast: Sea Stacks near Bandon, OR.

North California Coast: Giant Redwoods just south of the Oregon Border. closest town in Klamath, CA to the North and Trinidad, Mckinleyville, Arcata, & Eureka to the South.

Lassen Volcano Nat'l Park: Along and about a quarter of the way between Redding, CA and Reno, NV on California Hwy 44. I have not visited yet, but will be in June on my way to see the grandkids in Redmond, WA. Hot Springs among other attractions.

Yosemite Nat'l Park/Sequoia and Kings Canyon National Parks: Summer will be packed. Park entry Requires Reservations May 30th - Sept 30th. See the link.