We decided to visit the kids this summer, so drove from Portland, OR to Fargo, ND. We took the Southern Idaho route and ended up camping at the KOA just West of Yellowstone. Since Yellowstone and the Tetons are National Parks, be ready to pay up $25 to enter. We have a National Park annual pass which I strongly recommend if you're going to more than 2 parks in a year. It also gives you free entrance to National Monuments (not parking). I believe the West side of Yellowstone is the most popular and the busiest. However I hear it dies down a bit in late summer/early fall.
It's nothing but 2 lane roads all the way through. The running joke is - " There are 2 seasons at Yellowstone - Construction and Winter. " That said, there was a series of road paint trucks in one section. Another section they were fixing the road. Once you get past the painters, you stop to see something. Then you find they've gotten ahead of you again. The park is pretty awesome and it's BIG. If you want to take it all in, give yourself several days to enjoy it. We did the North Loop on one day and the South Loop on the next day and I feel that was rushed. So much territory to cover and the landscape changes all over the place.
You will probably see Bison and deer. Wolves are around but only in certain areas and we only saw 1 - from about 500 yards away, chewing on an animal carcass. Bring binoculars!! People love to stop on the road/side of the road. If someone THINKS they see something, they stop. Then everyone else behind them assumes something's there, so they stop. UGH! You'll see what I mean. Have a camera/phone handy. Bison do cross the road and walk up to your vehicle. It's cool to get pics or video when that happens.
I favored Old Faithful and that area because you can see the hot springs and all the colors and steam, etc. Plenty of shopping opportunity there too. Keep in mind that the park does not allow pets outside the parking lot. We took our dogs, let them potty in the parking lot area and left them in the truck with the AC running. That limited our sight seeing time because we were compelled to check on them often. It's best if you can leave them at the base camp BUT consider the round trip time. You'll most likely be gone all day. If you have the time, check out the park in sections and a few hours at a time. This way you won't be overwhelmed.
Grand Tetons isn't as big as Yellowstone but it's still pretty cool. You're more likely to see BEAR at the Tetons area. We got a glimpse of a black bear running through a parking area - rangers at the ready. Exciting! They have Moose out there too but they're hit/miss. It's really dumb luck when you run across animals. One area had a grizzly but a ranger blocked it off because some tourist decided it'd be neat to walk into the woods and follow it to get pictures. Come on people - wild animals will hurt you given the opportunity! Don't make that "selfie" the last thing you do.
If you're headed towards Custer and the Black Hills area, we loved checking out the small towns. Keep in mind this is the "old west" so it's all about cowboys and buffalo bill and indians. They've made a tourist trap out of the whole thing but it's neat to see the old town format and learn about the history. If you're a Harley Davidson fan, Sturgis is in that area. Go check out that town and visit the local motorcycle museum. Pretty cool old bikes in there. Get something for the wife as well. I got mine jewelry (Black Hills Gold). They're famous for it. Wall Drugs kinda falls into the small town tourist trap genre. It's small, lots of shops and overpriced fast food. Worth the trip if you're already headed that way but otherwise it's kind of by itself. Maybe if you're checking out the Badlands.
I'd definitely visit Mt. Rushmore but it had some construction going on when we were there. I wasn't too impressed with the Crazyhorse monument. Apparently it's still not done yet and someone's been at that for over 20 years now. Get a picture from the road/parking lot unless you just want to pay to be closer. Needles highway is cool - lots of twists and turns. DO NOT TAKE YOUR RV. Some of the tunnels are just big enough for a Pickup Truck. Be ready to see some wild animals in this area too. Bison, deer, mules, prairie dogs, etc. The wind cave elevator was broken so we couldn't do that trip but the rangers were nice.
One last thing. When you're in the Teton Area, go see the town of Jackson (Jackson Hole). Great SKI town - lots of upscale touristy stuff to see. Good food. Lots of shopping! I wish we could have spent more time there. Heading West out of Jackson was a beast - the pass had 10' degree or better hills. Towing my rig in a gasser was something in itself.
Good Luck and Have fun!!