I have traveled those routes, from Northern Michigan. First trip followed I-80, into Wyoming and then the south gate of yellowstone.
Second trip was US-2/ I-90 to the black hills of South Dakota. That was as close as I got to the east gate, then I turned north and went to Glacier and on into Canada.
There are a lot of things to see in South Dakota, Wyoming, Montana, both touristy and just of great natural beauty.
Where abouts in Michigan are you leaving from, and which yellowstone gate are you aiming for?