Here's Yellowstone's official site. Click on every inch of it because it gives a LOT of good information and gives all the campground closing dates.
YellowstoneAll national parks have an excellent site like this so take advantage of it. Definitely work in Grand Teton Nat'l Park which is right next door to Yellowstone. They are both completely different.
Grand Teton Nat'l ParkWe always stay IN the parks for the best overall experiences.
I might suggest you leave the 3rd or 4th week of August to get a head start and plan for at least a month for your trip. You have a long drive ahead of you and a lot of possibilities on what to see.