Since you have a month to get there and Paulj, I'm pretty sure he's concerned about snow that early. I would suggest heading to Utah and seeing that area at the best time...before it gets HOT. You could include the Grand Canyon...maybe even the N Rim which I prefer.
From UT depending on the weather I would go to Grand Lake CO for RMNP and then to Jackson Hole/Tetons. By then the south entrance into Yellowstone should be open. Yes, it adds 1000 miles to your trip but I can always go back to these areas.
If a thousand extra is an issue then I'd take I-70 past Grand Junction and then I-15 and I-84 which adds 200 extra miles.