We live 90 miles from the park, been there many times. Roads are narrow, and lots of bicycles to slow you down. Don't charge around a blind corner, you may find someone stopped in the middle of the road looking at a buffalo. I've seen a van and a motorcycle drive right into the river looking at animals. The motorcycle had a kid on the back. There was a Ranger in front of me so I let him take care of it.
In September, we took our Class C V-10 through Jackson Hole, Grand Teton Park, and Yellowstone Park over the Continental Divide three times, 8 to 9 thousand feet each time. Road is two lane with pull-outs but no problem V-10 hardly knew it was working.
Cody is my favorite place. The Buffalo Bill Museum does not take a back seat to the Smithsonian, I've been to both. The museum is huge and will take a day to see it.Check it out online. Cody has plenty of places to eat and stay, but it is a tourist town cut off in the winter so they have to make it in the summer, and about 60 miles from Yellowstone entrance at Fishing Bridge camp ground, the only full hookup camp ground in the park.
Know this, if you plan on taking an RV to a park camp ground, or West Yellowstone, make your reservations early. They fill up real fast. At Madison camp ground, the biggest one in the park, people are waiting to pull in as others leave in the morning.
I would not stay in Cody just to far out. Staying at a hotel in the park is very expensive. We don't stay in the park. I would suggest Grizzly camp ground in West Yellowstone, make reservation early very nice place. Take your time a lot to see and do in the park so you want to be close, or in the park.