In September, I would recommend stopping in either the SD or ND badland. IF ND, stop at the Roosevelt national park and also Glendive for the state park in the badlands. If coming through SD, stop at Devils tower in WY after stopping at the Black Hills. IN MT stop at Crow Agency/Garry Owen and tour the Custer battlefield.
When you get to Billings, take a side trip to Red Lodge and take your towed and drive up the mountain on the Beartooth HWY and then on to Cody on the Chief Joseph Hwy. Return via an alternate route or come back the same way, the scenery changes are interesting and all you have done is reverse the route.
Then either HWY 89 to Gardner from Livingston or 191 from Bozeman, although I usually recommend coming to Belgrade and fueling before heading to Yellowstone, cheaper than near the park. 191 is very scenic and follows the Gallatin River a good part of the way.
A comment about Bozeman, the Museum of the Rockies is there and they have great dinosaur exhibits. The curator there was the technical advisor on the Jurassic park movies. He has some "interesting" theories and approaches.
There are other more southern routes but I am not familiar with them, one at Sheridan, another at Buffalo and the third at Casper WY.
Cody WY is an interesting but small town. The museum there is great.
Whatever you do, do not miss the Grand Tetons.
If coming in early September, I recommend coming through Billings and taking the Beartooth HWY, if open. September is our season for change so be prepared for cold, teens or lower, and snow. If you have snow, it will be gone in a day or so. OR, you miss it all, bad weather that is.
BTW, September and October are my personal favorite months in MT, not too hot nor too cold, autumn seasonal changes and lots of wild life coming down from the mountains.
Have a great trip.