If you plan on fishing, you should know that Grand Teton Ntl Park requires a Wyoming fishing license, while Yellowstone requires you to purchase their separate park-only license.
The Flagg Ranch suggestion is a good one. In no-mans land between the two parks, it is not as scenic, but it does make it convenient to visit both parks. Leeks Marina a little ways to south has great pizza.
Around Phelps Lake in Grand Teton is a good hike and has decent fishing near the inlet. String Lake is also fun for an afternoon picnic-- but the fishing sucks. The Gros Ventre river is good for road accessible fishing both in the Park and the National Forest to the east.
