The National Parks are a good start. Here are my 5 picks:
I'd put Crater Lake up near the top of the list but keep in mind it's only completely open a couple months out of the year. We went there and the rim road was closed, there were piles of snow 10' high. This was in June!
Dante's View in Death Valley is spectacular, a one-of-a-kind experience.
Grand Canyon - try the North Rim instead of the more popular South Rim. Where the South Rim is more desert-like the North Rim, 1000' higher, is heavily forested and gorgeous.
Glacier/Waterton Lakes NPs (US and Canada respectively) are incredibly beautiful and shouldn't be missed.
Rocky Mountain National Park, up Fall River road and then over the pass to the West side (where you're likely to see moose -we've seen them every time we go there) and then back over Trail Ridge Road, the highest continuous paved road in North America (about 12,200') for unbelievable scenery and wildlife.
There are others I'd recommend - In Utah alone there's White Rim Road in Canyonlands, Valley of the Gods, Capitol Reef (going there in a couple of days!), Zion and Bryce Canyon.
BTW The best time to see Bryce in the late spring when there's still snow on the ground - the contrast between the snow and the red pinnacles is stunning, especially at sunrise.