For Traveling from the East Entrance to the West Entrance, Cody, WY to West Yellowstone, MT, it's a little backwards than what I would suggest. Just because you will visit one of the pinnacle sites firt, rather than last, and it's nice to finish with dessert.
From Cody to Fishing Bridge Junction is about 80 miles and you will only be going about 40mph so it's going to take a couple hours. You will follow the Shoshone (SHOW SHOWn(EE)) river out of Cody and into National Forest. In the forest you wind along the river, but evenutually the river heads north and you climb west toward Sylvan Pass in some high mountains.
As you top Sylvan Pass you will be able to see into Yellowstone which is mostly a vast plateau. Decending down you might want to pull over twice. First because it looks like a good picture spot, again because it looks like a better picture spot. With a good eye, out across Yellowstone you can see the Grand Tetons pointing upward. Also to the west is the pink Lake Hotel on the shore of Yellowstone Lake. At Fishing Bridge you can see Cut Throat Trout in the river, it's a spawning area now closed to fishing. Make a right at the junction toward Canyon Village.
Here you follow the Yellowstone River. A few good picnic spots along here. On the left will be the Mud Volcano thermal area, the first mandatory stop. Although there are no geysers here I consider this the best thermal area. It's very unique, dynamic and rememberable. It has a short loop to walk and takes about 30 minutes (I skip looking at the couple features across the street). As you follow the Yellowstone River north you cross Hayden Valley and approache the Canyon of the Yellowstone. You really won't see the canyon from the main road.
Make a right off the main road toward Artist Point. But go into the parking at the first left to Uncle Tom's trail. Uncle Tom's trail will take about an hour. It drops into the canyon. Descends semi steeply on switch backs and then has hundreds of stairs. This is the only way to really get down into the canyon. You might opt to continue to Arist Point, if you do this I'd skip the one way road, after visiting Canyon Village.
After this go back to the main road and north to Canyon Village. On the right the Brink of the Upper Falls is a must see, with about 5 minutes of walking it takes you very close to the brink of the Upper Falls. You can watch the large Yellowstone river get channeled into a rushing, blue slot through the rocks and then turn to white as it drops into the canyon. After continue to Canyon Village.
At the four way stop make a right to Canyon Vilage, just to the right. Entering here on the right is a very good museum in the Visitor Center. Strait is a store and grill, and ice cream in the Hamiltons. On the left is Canyon Lodge which has cafeteria style food.
After visiting Canyon Village exit going right onto a one way road. To the left is Inspiration Point(optional) but it is quick(hop in and out, no walking) gives a specatular view of the Canyon. Back on the one way road there is one other great stop. The Brink of the Upper Falls. While the Lower Falls Brink trail has a walk of about 20 minutes, it let's you view the taller waterfall that goes to the bottom of the canyon. The one way road will take you back onto the main road where you go right to get to the four way stop at Canyon Junction.
Now it may be possible if you got a jump on the day to go strait, north through this intersection, up to Dunraven Pass. Parking there you can take a difficult hike up Mount Washburn. It will take about 3 hours. From the top you can see a large portion of the park. Often on this hike you will encounter some of the more rare Yellowstone Wildlife.
If time does not permit a long hike go West at the 4 way stop toward Norris Junction. At Norris is another 4 way stop where you will want to make a left toward Madison Junction. In the Gibbon River Canyon are some small thermal features and the cascading Gibbon Falls, I would just view these from the car as you drive by. You will descend down to a slow speed zone at Madison Junction. Later from this Junction you will exit the park to West Yellowstone, about 25 minute from the Juntion to your hotel. But to continue touring Yellowstone go strait.
On the right just after you start to climb a hill from the river is a one way road, Firehole Canyon Drive. For an adventurous family I would stop either now or later if time permits (you will back track this entrie section to get back to Madison Junction), but if it's not early afternoon do it now. Along the Firehole Canyon Drive are two points of interest. Firehole Falls, might warrant a stop, and the must stop, Firehole Swimming Hole.
The Firehole River is fed by many geyser basins. So it's warmed up, to what I would call unheated swimming pool temperature. Better than the crisp streams that flow off the rest of the Rocky Mountains. As you descend the stairs down to the river to the right is a calm swimming area. On the left are the rapids which you can jump into and float through a narrow gap, the water fizzes. Strait are some cliffs, while it says no jumping, I've fallen into select pools, hundreds of times, from those cliffs.
Continuing on south you come back to the main road, make a right to keep going south, toward Old Faithful. Stop at Midway Geyser Basin. This takes about 30 minutes to walk through but has some of the best pools to look at. Then back in the car go right toward Old Faithful.
Off ramp to Old Faithful. So normally I suggest staying on the Old Faithful access road all the way and park in front of the OF Lodge for the closest and best parking. But with a camper I would take the first left. It's a little risky but see if you can park in front of the Old Faithfull Inn, or behind the Hamilton Store, parking can get full around here.
In Old Faithful see the Visitor Center and the OF Inn, both will have an estimate of the next eruption of Old Faithful. If you have little time look around the buidling, then go out to the boarwalk. If you are lucky in the visitor center look at the eruptions predictions for geysers nearby. Some specatacular geysers like Castle and Beehive are about 10 minutes down the old road(the paved trail between old faithful and the buildings) so if your timing is right, they are scheduled to erupt, or you have a lot of time to wait on Old Faithful you can wander down there and then loop back on the board walk through geyser hill in the Upper Geyser Basin.
Ideally you are here around early evening and can eat in the camper or at the Grill in the Hamiltons Store, the Cafeteria in the Old Faithful Lodge, the Restaurant(reservations) in the Inn, or the restaurant in the Snow Lodge at the back of the parking lot.
On your way back to Madison Junction, depending on time there are two great things to do around dusk. Walk the Lower Geyser Basin or take the one way Firehole Lake Drive. You can get great sunset photos.
A couple notes. You really have to get a great start on the day. In the parks there is nothing going on after dark. You can't see the sites.
You can have a few things slow you down. Animals near the road can cause traffic jams that can delay you 15 to 45 minutes. Construction can stop traffic for 15 or 45 minutes.
Gas is really expensive around Yellowstone. Airport Prices. Even West Yellowstone will have higher than normal prices. So it's good to top off in Cody, Wy so you don't need to fill until back near the freeway.
Want to take me along? 
But for a good driver, no problem.
) take your time down the coast and make lots of Seal watching stops as you go. Stop in at one of the many Santas along the way.....Santa Barbara for us this year......then take the plunge into LA.....The drive to LA could very well be the highlight of your trip as your photographs will remind you......just an amazing coast to drive!
