I'm not sure where you're going to end up, but I took the Amtrak from San Francisco to Seattle and the ride is incredibly gorgeous. Plus, San Francisco and Seattle are both excellent places to visit.
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I highly recommend Portland Or, especially in summer, since it's never super hot and it's always beautiful and fun.
The train ride from Denver to Reno is legitimately supposed to be the most beautiful in the country! So I'd do whatever you can to ensure that's a part of your journey!!
We just spent a week in Flagstaff, AZ and had a great time. We spent our time waking around downtown trying a bunch of excellent restaurants and local coffee joints. It’s just a cool little town.
One of our days we made the couple hour drive to the Grand Canyon, which is obviously highly recommended. Bonus: if the Govt is still shutdown, you’ll probably get into the park free like we did.
It looks like Amtrak’s Southwest Chief will take you the whole way.
Okay I think you should take the coming-back weather into account, driving in your rental. In summer, the less desert driving the better, whereas it's easy in a train, so maybe the Southwest Chief that goes near the Grand Canyon. It ends in Los Angeles at Union Station, where you could catch the Coast Starlight and head north, or go Metro on the Expo Line to Santa Monica, to stick your toes in the Pacific and/or rent a car.
Then come back over the Rockies, or if you have time drive north along the coast first and come back across the top.
The Sequoia National Park (Kings Canyon visitor center) is about a 5 hour drive from Santa Monica if traffic cooperates, though it never does in LA. It's uniquely beautiful imho, worth it even though it's out of your way for the rest of your trip.