It's a drive that could easily take you 4 weeks if you stop everywhere! Not knowing how much time you have available, here's a few things I would recommend. NOTE, bring water, food, and fuel. The distances in the west are 10x more than you will expect. VASTNESS.
I90 gives you the opportunity to see the Badlands and Mt. Rushmore. But might be too far out of your way.
I80 is going to be flat fields from Iowa to Denver. There are some car and military museums, but I forget where. I think Nebraska.
I70 is going to be flat fields between Denver and Iowa/Missouri.
Here's the trip I would take based on my experiences.
*North to I80, then west, if you have the time. See the Badlands and Mt. Rushmore. (Otherwise your drive across Kansas and Nebraska will be very dull).
*From Wyoming, S. to Denver, then west on I70 to Moab Utah. There, check out Delicate Arch, Canyonlands, and Needles National Parks.
* Alternately, shoot west to Jackson Wyoming. The Grand Tetons and Yellowstone are a must see. Then shoot directly to Reno shortest route is thru SE Idaho. Hit Moab on your way back after Vegas, you'll have to backtrack a bit.
* From Moab area, North thru Salt Lake City, then West to Reno. Reno has a cool car museum. You'll be very close to Lake Tahoe California, Sierras, which is a breathtaking area worth checking out, along with Truckee the infamous Donner Pass. Beware of California very strict gun laws if you have guns with you.
* From Reno, get yourself to Vegas and the Hoover Dam as you wanted.
* From S. Nevada, scoot over to SW Utah and see Bryce Canyon National Park and Zion National Park. Both are amazing.
* From S. Utah, scoot South to N. Arizona to see the N. Rim of the Grand Canyon and Lake Powell, Arizona. Make sure to see famous Buffalo Canyon.
These are must see.
* (Might be out of the way but 4 Corners is unique, and Mesa Verde in S. Colorado is very interesting. Probably too far away)
* From NE Arizona, it's a shot SE to El Paso. From there I'm not familiar with anything en route to Missouri. Perhaps the Ozarks of Eastern Oklahoma?
I love the USA and road trips. Plan plenty of time and bring supplies.