West Yellowstone: Grizzly RV Resort in West Yellowstone and about 1/2 mile from the west entrance.
A bit expensive but worth it.
Lots of playgrounds for the kids also and lots to eat and buy junk in town instead of within the park itself.
Park takes at least 3 to 4 days to see everything the first time.
Make reservations for campground as soon as you know when you will be there.
Fills up fast and you show up with no reservations sorry no luck.
You can drive your RV into the Yellowstone National park on the west entrance and drive to the south entrance which is where you will hit the Grand Tetons. I have stayed in Colter bay Campground in Grand Teton NP.
Then continue south to Jackson Hole then south to Utah and Las Vegas the Arizona and the Grand Canyon.
Suggestion for Arizona and the Grand Canyon is all I can suggest:
Grand Canyon Railroad RV park is located off of I-40 just south of the Grand Canyon and located in Williams Arizona.
Best campground with everything you need. Full hookup, Wifi , Cable TV etc.
They have a really nice full service restaurant that is in conjunction with the hotel there.
The reason the call it the Railroad RV Resort is because they have the steam and diesel trains that can take you to the south rim of the Grand Canyon. They have bus tours at the Grand Canyon as well as lunch and you return to the end of the line in the late afternoon and you walk back to your RV.
For first timers this is the only way to see everything and ride on a real old steam engine railroad.
Prices vary but all car are air conditioned and I would think about taking the observation car going in so you can see all around before you get to the Grand Canyon.
Each direction is about 30 miles I think and takes about an hour each direction.
At the end of the line at the Grand Canyon you can jump onto a large bus that will take you to all the look out and stops and lets you take pictures.
After all that it takes you back to the Grand Canyon village and you can get a box lunch (part of the tour bus) or get something on your own.
The train then leaves about 3;30 PM or so for the relaxing trip back to the campground.
The train has drinks and snakes and gets robbed by the bad guys on the way back. Kids love that. The bad guys ride horse along the train route just like in the movies then come aboard and you can take pictures of them with the kids.
Here is the link to the campground and you can also make reservations for the train all in the same place. (by Phone)
If you have any other questions about Arizona ask away.
Thanks all I can tell you at this time.
Enjoy your vacation.