I suggest taking I-90W from Rapid City into Wyoming. Just passed Sheridan is the exit for Ranchester(US 14W), that will take you all the way into Cody. If my memory is correct, it is exit number 8 or 9...but please don't hold me to that. The 2nd exit for Sheridan has a great ExxonMobil station with real easy access for fuel. There is also a McDonalds right next to it with a fair sized parking lot. US 14 will also take you into Yellowstone from Cody, but I have only traveled as far as Cody on 14.
As for coming out of Yellowstone, I would say grab US20 South and merge onto I-15 in Pocatello, Idaho. Then you can stay on 15 all the way into Salt Lake City, Utah. I left Spokane, WA last August and went to the fall rally in Indiana via Salt Lake City. I took I-15 out of Montana all the way down to SLC. By the time you get to Pocatello you are pretty much out of the "mountains". I am confident your wife will not have any trouble with I-15 into Salt Lake City. I cannot vouch for US20.
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