Avoid 14A and the Beartooth is out of the question imho. Those would be my recommendations. I live in Helena MT. I would guess it would be where you want to end up that would dictate. If you want to go toward the North and end up in the Tetons, I'd do the east entrance through Cody. Come from Custer on 16 to 90. We do it once or twice a year.
From the South hitting the Tetons first head down to Riverton and 26 in to Dubois. There is a nice but expensive place in between Tetons and Yellowstone that used to be called Flagg Ranch, not sure if it still is. It is a decent place to base if you want to be central to both areas. Makes for a long trek north into the park for day trips. It has full hook ups. Good luck getting into Fishing Bridge as you are likely late within a half hour of it opening online.
And in reality, we go nowhere near Yellowstone during the season. It is one massive parking lot. The best times are the two weeks before kids get out of school and two weeks after they go back or best yet, Winter and stay in a motel.