Hmmm. Interesting topic. I'll try to list some of my favorites.

Walking across the Grand Canyon. At times, you are alone on the trail, and wonderous sights abound. Going part way down is a poor substitute, but better than nothing. Be sure you stop along the way. Talk to people and share your experiences. One never forgets.

Climbing at least partway up Mt. Rainier. Mt. St. Helens must surely be a pilgrimage site, though I have never been there.

The Four Corners area, just to say you'd been there.

Head out into the really wild country around Moab, Utah ~~ Arches, Canyonlands, etc., but make sure you remember the way back, because if you don't, it is likely that no one will ever find you.

Death Valley. Death Valley Scottys, of course.

Yellowstone. Go to the less traveled areas. There are some.

And there are the standard things made or altered by man ~~ Statue of Liberty, Lincoln's Tomb and home, Liberty Bell, multiple places in Washington, D.C., Mt. Rushmore, on and on.

I think you'd even have to add Mountain View, Arkansas. Some people do make that pilgrimage every year ~~ some stay a few days, some all summer, year after year. I know of some New Zealanders who come every couple of years. They wouldn't miss it on their trips to the US.