We have traveled to Yellowstone and Grand Teton several times from Southern California. The Route you have outlined should not present any significantly steep grades. Once you are inside the boundaries of Yellowstone and/or Grand Teton NP, the roads are well maintained and not very steep, despite the fact that you may cross the Continental Divide numerous tomes on your journey. The biggest safety concern is to watch out for abrupt traffic stoppages when tourists stop and park in the middle of the road to try to get a glimpse of a Bison or Bear or Wolf along the side of the road!! We have actually seen families exit their vehicles with their children and try to get near a Momma Bear and her cubs for a photo!!! Do not approach any wild animals on your journey, no matter how serene and peaceful the scene might seem.
You're going to love Colter Bay.