We just returned from a 19 day trip from Tucson to Yellowstone and the Grand Tetons and while our trip was just my bride and I in a 24' Class C here are few thoughts to share. Since you will be in Cody do not miss the Buffalo Bill Center of the West but try an allow at least an hour or two to explore. You could spend more time but a couple of hours will give you a great look at this amazing display.
We drove from Cody across the Yellowstone to West Yellowstone where we stayed at the Yellowstone Grizzly RV Park {about one mile from the west gate of the park}. This is a very nice RV Park that I highly recommend.
Access to Yellowstone from here is excellent especially as you have Old Faithful near the top of your list. The ride to OF is awesome with lots of thermal areas too explore along the way. Buffalo and Elk by the hundreds typically graze along most of the route.
We spent 4 - 6+ hours 6 days out of 7 exploring the parks in our coach on the cold days and on my motorcycle when it warmed up and covered nearly 700 miles. We did not see all of Yellowstone but we did see most of it. Do not miss the Grand canyon of Yellowstone {see it from both sides of the canyon}. The waterfalls will take your breath away, one is 308' high as the Yellowstone river roars through the narrow canyon.
The Mud Volcano is another great stop with lots to see. Leaving Old Faithful continue east and then north along the west shore of the lake {going towards Fishing Bridge {the Grand Canyon is just north of FB}. If you have the time a trip up to mammoth Springs near the north entrance is another great ride with lots to see when you arrive.
Not sure how you accommodate 7 in one vehicle and for how long that remains fun but Yellowstone is so amazing I have no doubt you will all have a blast.
We drove south out of the park and also stayed at Firesides Buffalo Valley RV Park and loved it. The views of the surrounding Grand Teton's are incredible. The restrooms and laundry facilities were immaculate and they have a small store and gas station on the property. A drive into Jackson is scenic beyond belief and it is a fun town to visit just 40 miles down the road from the RV Park. The Jackson Lake area is another great area to explore.
It is impossible to pick a favorite photo but this one works about as well as any. The smile on my bride's face pretty much says it all. This is about halfway to Jackson from the RV park at the Glacier View turn out:

Different trips for different folks but Yellowstone and Grand Teton are about as good as RV'ing gets. Hope this helps... Enjoy!
:B