In your specific case, I’ll agree with others on renting a vehicle, for all the good reasons already offered. Of the other options mentioned, I’d recommend against the idea of transporting a car for someone else. That works if you are just trying to get there - these arrangements are explicitly for going directly from point A to point B without stopping to sightsee, dawdle, and smell the roses, all of which you are hopefully planning to do. I’ve done the transporter thing a few times - it was useful at the time, but not for exploring.
Back to renting: If you can possibly manage the cost, you might consider renting a very small motor home, hardly bigger than a pickup truck, so that you can wander off the beaten track and camp in established campgrounds in National Parks and other areas now and then. There are probably several companies which offer these, do a google search for RV rentals. You could perhaps rent a small car for the eastern states, then pick up the mini RV once you reach the wide open spaces.
Take as much time as you possibly can for this great adventure. Once you cross the Mississippi, the landscape and open space changes everything. Tons and tons of great natural places, but I find that exploring the agricultural heartland is a rewarding experience too, it’s so much different than the urbanized and compact eastern states.
Because of mountainous terrain where snow lingers into summer, keep in mind that higher elevation highways are subject to travel restrictions much later into the spring than you might imagine.
My top recommendations for parks, in no particular order, include Yellowstone (WY), Glacier (MT), Arches (UT), Grand Canyon (AZ), White Sands (NM), Rocky MT (CO), Sequoia (CA), Yosemite (CA), and the state parks in redwood country in northern CA. In general, southern Utah and the coast from northern California to Washington are especially beautiful. But that only scratches the surface of great places in the American west. Take time to explore small towns in Kansas and Nebraska, drive out into serious farm country, take two lane roads, all of which will result in great adventures. Enjoy!
And please…send reports from the road!