[Josh-PA]

We've been a few times, for me it is the most amazing place in the world that I've ever been to (and I've been a lot of placed). You need to get off the road in order to really feel the majesty and scale. If you do, you will totally feel like you are no longer the top of the food chain and the master of your domain, it is wild. We usually go the week our school lets out (my son is a senior in high school this year, so he was a sophomore last time we went). The early summer is nice because the weather is still cool, crowds haven't spiked yet and the animals mostly have babies.

There is stat that something like 95% of Yellowstone visitors never get more than 200 yards off the road.

My suggestions (PM me if you want details, I have old itineraries, etc):
Fly into Jackson Hole and stay at a nice hotel in Jackson day 1 / 2: we like the Cloudveil but we are Marriott loyalists. Depending what time you get in, you should be able to spend an afternoon in the Tetons hiking a bit. You may want to hike behind the visitor center at the Snake river towards evening, there will be a very good chance you'll see a moose (pic below of the mama/baby we saw). Go to Hatch for some Mexican food and Tequila drinks for dinner.

Next Day start early and do the hike to Delta Lake (a cut off from amphitheater lake trail). It is a long, climbing hike but you wind up at a lake at the base of the treeless portion of the Grand Teton peek. On the way back down detour onto Garnet Cyn trail a bit, amazing views. You should 100% see pronghorn at the base of the trail / on the drive in. They are everywhere here.

Day 2 check out early and drive through the Tetons up to Yellowstone. Swing by Jenny lake to see it (underwhelming) and grab lunch up by Colter bay and overlook the park from the North. Head into Yellowstone. Stop by Old Faithful and check that box. Be sure to follow the boardwalks back to the left - that's where all the pretty hotsprings (like morning glory) are. We stayed 2 nights at the Lake Yellowstone hotel area and worked the southern half of the park from there.

Day 3 goto the Grand Canyon of Yellowstone and plan some hiking from various trailheads (not just Artist Point). If you head that way early you'll see a lot of wildlife along Hayden Valley. Be sure to stop along the way to check out Mud Volcanos,etc. On the way back definitely plan some time to watch Hayden Valley for wolves / bear, the elk and bison herds, etc. If you like to fish, book some time with a boat rental and nab a bunch of invasive lake and rainbow trout.

Next Day, drive the opposite direction from the lake and go to Grand Prismatic (see pic below and notice all the tourists on the main boardwalk). The boardwalk will be MOBBED with people. Look up the Grand Prismatic overlook hike and take that up to a hill overlooking the spring from behind. Much more impressive views and less people. The Fairy Falls hike there is pretty cool too.
I'd suggest from Grand Prismatic head up to Mammoth for the remainder of your stay in the park and use that as a base for the North side. If you want non-park food drive out to West Yellowstone. It's very touristy but has a ton of average restaurants.

If you want a break the next day drive out of the park to Gardiner. Grab some Elk Chili at Wonderland. Gardiner and North has some fun horseback riding / whitewater type of excursions that can be a nice break from heavy hiking. The drive to Gardiner from Mammoth is your best place to see bighorn sheep.

From Mammoth Hotel - one of our favorite hikes was the beaver pond trail loop. We had quite a grizzly bear experience there...
You can spend an easy half day walking the Mammoth Hot Springs and area behind them, it's pretty neat.
Definitely spend a day out in Lamar Valley. Best wildlife viewing in the park. If you go all the way out 212 to Barronette Peak you should be able to see mountain goat.

Head back to Jackson and relax before heading home.