Don't go out of your way. You can easily drive right through. Even if it might be slower it will give you an added chance to see more critters. We drove through one time when the rangers were on horseback herding the bison to another part of the park. They rode right next to us which was neat and it was fun watching them wrangle in a wandering bison.
As mentioned, definitely include Grand Teton Nat'l Park in your stay. They're right next to each other. If you hit the Teton first from the south it's an easy shot north through Yellowstone to West Yellowstone. The Tetons have a couple huge 300-site dry campgrounds and we've always gotten a spot without reservations. Gros Ventre is our favorite. There also a full hookup in the park called Colter RV.
Have a great trip! We've been to both often and never tire of them.