I assume you'll be coming across I 80 through Wy at some point, from there I usually take US 191 Straight into Jackson Wy, which is just outside of Grand Teton. The last town you'll hit before getting to your exit would be Rock Springs. This has the last Walmart super you'll see, and the cheapest fuel until you get to Jackson, fuel prices only increase as you get closer to the park.
One very nice dry campground is Camp Kennedy at Gros Ventre just 2 miles outside of Grand Teton. There are several campgrounds inside the parks, but each park has only 1 campground with hookups. Coulter Bay in Grand Teton, and Fishing Bridge inside Yellowstone. Both are great, but Coulter bay is nicer. It's much more wooded, and the village has a lot to offer. You may see Grizzlies at Fishing bridge, we had one in our campground, so make sure you follow the regulations concerning food. It really takes a solid week to get everything in, and appreciate it. Our big trip this year is Olympic, Ranier, and Redwoods, but I have to go to Yellowstone at least every other year. There's no place like it.