3 weeks roadtrip in the west and rockies

[kanadianDri3]

Hi, everyone!

With 2 friends, we are planning a road trip across the Rockies and down the west coast.
We will be starting in Las Vegas the 1st of June. Find our way to Denver. Drive north to Theodore Roosevelt National Park. Turn west to Seattle (and stop by Yellow Stone) and cruise south to southern California.

We have to be in San Fransisco the 20th or 21st, so the southern part California is optional (from Yosemite, go to SF directly)

I realize that we won't have time to do everything. So I'm asking the community what is worth a stop of 2 or 3 days (to hike and explore). We only have 3 weeks unfortunately so I was hoping to do a few sites in the same day (like drive thru monument valley or stop at mount rushmore an hour).
Any tips and advice are more than welcome.

We were planning on camping in National Forests on the way. Except in some places like Yellow Stone where we need to take a reservation, is this possible?

Thanks a lot!