We took a couple of weeks, but drove ~2,500 miles.
You're pretty reliant on using the passes through the mountains to get to the most stunning areas, so winter is probably not best as the high passes are closed to normal traffic - we were there in the summer, but even so, there was a snow 'event' when we were travelling on the 'Going to the Sun' Highway:



I took this photo the previous day:



I can't remember the car we used - but we could have done with something different when I decided to go off-piste in the sandy tracks through the Missouri Breaks!

Just remembered, I did a thread about this trip yonks ago.

Arizona is good too - and November is a good time for that. We've been a couple of times, first to do the Grand Canyon type touristy stuff, then to concentrate a bit more on the back-roads and little places.