We stayed at Madison campground as it seemed to be the most centrally located for what we wanted to see. Expect a lot of backtracking, as there are no direct routes to all the points of interest in the park. Madison was at a lower elevation so it was warmer than some of the other cg's. There aren't any showers there, but they do have flush toilets. No power anywhere but the cg at Fishing Bridge, and they don't allow tents or pop ups there. It's a good idea to make reservations as the most popular cg's fill up fast. If you have a large rig, good luck because there aren't many large sites. Ours was typical.

The only way we could get our rig on the site was to back the truck up almost to the slide dinette on the back side of the trailer, and hang the back bunk over the grass. All wheels must stay on the pavement, and they're not kidding about that either. As you can see, the sites are very close to the road. (The dogs are on six-foot leashes.)
If you don't have a hard side camping rig, plan to keep ALL food and toiletries in your vehicle. They will give you a long list of things you can't have in a tent or pop up when you check in and they're not kidding. There are bear boxes but they're not very plentiful and not on the sites that we could see. We kept everything in the back of the truck, which has a hard top on the box (Chevy Avalanche.)
Expect gridlock everywhere you go, so allow plenty of time to get anywhere. Also, the worst drivers on the planet will be there, and a whole bunch of them are in rented RVs they don't know how to drive. Among them are a bunch of people who aren't from the US and may not be familiar with our driving laws and/or able to read signs posted in English. It's frustrating, but try to remember they're guests in our country and treat them the way we'd like to be treated if we were to visit theirs. Yellowstone really is unique in all the world, and it's no wonder so many people come from abroad to visit there.
Pack your biggest bag of patience and dip into it often.
We spent one night at Mammoth campground the night before we left the park. Beautiful! But too far from most sights to be practical for the entire stay.
West Yellowstone, Montana, is near the west entrance and not a bad drive from Madison cg, maybe fourteen miles. We used the laundromat there, bought groceries (two stores, both WAY small and WAY expensive), used the library computer, and also bought way too many tee shirts at a tee shirt place there. It's by far cheaper than shirts in the park, and the staff was highly entertaining as well. The name of the place is Yellowstone Outlet.
Groceries were also way cheaper in West YS than in the park and so was ice. We bought a deep cycle battery at the NAPA store there and the price was not out of line.
The entrance fee to YS also covers Grand Teton.
All the national parks have excellent websites that give a ton of details. There are also webcams at places like Old Faithful.
I read on the camping forum I frequent that showers cost $2 for, I think, six minutes. We were glad we had added a shower to our pop up. It came in handy, worked great, and saved us money.
Don't leave your specially-purchased, 4-gig, $10 apiece camera cards at home on the headboard of your bed.

We ended up paying $25 for a 2-gig card in the park, and were grateful to get it at any price.
We used a car charger that plugs into the cigarette lighter in the car and has a standard two-slot jack like a wall outlet at home. It worked great for charging small stuff (cell phone, Nintendo DS, etc) while we were on the road, which was 8-10 hours per day inside the park. Mostly we used it to charge camera batteries so we always had fresh ones. They are available in the park at twice the price you can get them at home.
If you're coming in through Cody, Wyoming, there is a Super Walmart that's handy for stocking up on groceries on the way to the park.
Come prepared for very changeable weather, and dress in layers everywhere you go. It can be sunny and warm one minute and windy as heck and cold the next. Literally! Be prepared for chilly weather overnight, too. We didn't run our furnace in our trailer except to take the chill off a couple mornings. The coldest it got in the trailer was 38, so not bad. We always have extra comforters so were toasty in our bunks.
If you're planning to run gas appliances at the park, keep in mind you'll be at elevation which can affect things. Our furnace was reluctant to light. Our water heater lit just fine, but wouldn't stay lit unless we left the door open. Our portable gas grill wasn't as efficient as usual. Etc.
Speed limits on the park roads is a max of 45, but you'll frequently be stuck in traffic or held up by bison jams. Our eight-cylinder gas hog truck got twenty mpg while we were in the parks. We drove over 800 miles during the six days we were there, so plan for extra gas money just for getting around the park.
If you plan to buy lots of souvenirs in the park stores, buy the membership in the Yellowstone Association as soon as you get there. It'll save you money in the long run.
If you bring huskies along and/or have a license plate from a state that people recognize as having a world-class wilderness destination, add extra time for people from all over the world to pet the huskies, have their pictures taken with them, or chat you up about your state. For us it was one of the most fun parts of our trip. We also learned we need to carry along tourism info from our town to avoid writer's cramp from giving people tons of URLs so they can look up the info they quiz us about.
Always keep a camera handy.
That's all I can think of offhand. If you have more specific questions, fire away.