Wise to go to Grizzly first and park the 5er. Enter through the West gate early. After about 10AM, the line will be long due to early arrivals to the area buying their NP pass at the gate. If you buy a National Parks pass before leaving home, (I bought mine at an American Express office in Cincinnati, OH), there won't be so many other visitors buying that at the entrance. I barely had to slow down at the gate each day. I chose to visit one of the four quadrants in YNP each day for 4 days. There will be buffalo jams and gawkers making for slow travel every day. (short season).
Do not try to drive through the herd. Buffalo have thick sculls and small brains. I drove past a lady with a severely dented door complaining to a Ranger. It didn't do any good.
Crossing the Bighorns is a "Challenge". I drove 14A in my Jeep Wrangler "toad" and I swear I could see my own rear license plate through the front windshield. (Slight exaggeration). 16 is the easiest, but even it has places to pull out to check your brakes. Do it. If you boil the brake fluid by riding the brakes, you become a sled, and you can't drag your feet to stop.
Finally, head south to the Tetons after YNP. They're next to each other but completely different. Be sure to take a small detour onto the Jenny Lake loop road. There's a pullout that holds about 6 cars. Park and walk over to the edge, sit on a rock and feel your high blood pressure drop about 40 points. There has never been a gasoline boat motor on it, so you can see the bottom. Probably the most relaxed I've ever been while awake.
And if you don't take a camera, you'll be sorry. Its been years, but I still look at them from time to time. Takes me back, and its cheaper than gasoline.