Gene,
What route do you plan to take to Yellowstone?? If I-94 to I-90 and then on to Yellowstone, come west all the way to Belgrade MT and then go south on MT 85 and US 191 to Yellowstone. Then do Yellowstone and the Tetons and stop on Cody on the way out. Do the Cody to Red Lodge and return from Cody. The Best of both worlds. If you can, bring the car as it will be a lot easier to do Yellowstone, Tetons, etc. If not you can rent but if you do get advance reservations ad they will be expensive as there are lots of tourists that fly in to Bozeman, Billings, West Yellowstone, etc. and cars are in high demand most of the year.
There is 1 pass and a steep hill between Billings and Belgrade/Bozeman. The hill is near Columbus MT and it has a rest stop near there. The pass is between Livingston and Bozeman and the total altitude change is about 1500' with the last 1/2 mile the steepest going west. There are several climbs along the way and in the parks but nothing that should stress either the driver or vehicle. Watch the temperature and take them slowly and you will do fine. IF you get to Belgrade, we are in the phone book and would enjoy meeting you IF we are home, we have some family events this summer elsewhere.
BTW, there is a world class museum in Bozeman if anyone is interested in dinosaurs