Spring can be an iffy adventure heading out west. I would shoot for leaving mid to late August. Kids will be, or are, in school in many areas so campgrounds shouldn't be full. The weather is usually pretty nice if not warm at that time of the year. Labor Day weekend might be busy so you could always try to target someplace by reserving a site for that time period. Otherwise I don't think you'll need reservations anywhere.
We did a Michigan to New Mexico-Colorado-Wyoming trip a few years ago in 3 weeks. I did pull our 5th wheel on Hwy 550 from Durango to Ouray and it was beautiful, but exciting with all the mountain driving. We did some long pulls, especially on the way out & back, but had some nice stays, usually 2-3 days in most locations we wanted to see. Grand Teton Nat'l Park was a week though. If you're not into long daily pulls I'd stretch out your plans to at least 4 weeks. Northern route first and come back southern route.
I've visited Mt. Rushmore and can't say it was all that exciting. Grand Teton & Mesa Verde are two of my favorites. Utah has plenty of excellent national parks too.
And just for fun here's a youtube music video for all those folks telling you that you'll just looking at the scenery going by your window due to the distance and time factor. Be sure your sound is on. The song is Looking At The World Through My Windshield by Commander Cody. It's cute and the scenery isn't bad.
https://www.youtube.com/watch?v=l3ZhD6Ei1bM