Shenandoah and the Great Smokey mountains.

Drive the blue ridge parkway. it's about 4 days with plenty of great sights off the side of the road.

Breaks interstate Park between WV and KY is pretty nice. I also very, very highly recommend smoky mountains, I live in the region and have gone every year. Ik you said no cities but Gatlinburg area can be nice to check out after the trip if you wanna see Smoky Mountain Knife Works and all the small businesses in the rural parts of the mountains.

If you're willing to go as far as Kentucky or West Virginia anyway, you should consider the Red River Gorge area in Daniel Boone in KY, or the Spruce Knob/Seneca Rocks area in WV. Neither are national parks -- they're both national forests.

Both will be considerably less touristy and less crowded than (at least the popular) national parks, and you don't have to pay just to get in, either. These two areas have some of the most bodacious geology on display on the East Coast, in my opinion, and if you're into that sort of thing it's well worth checking out.

The heyday of the Seneca Rocks region seems to have passed and getting accommodation there that's not camping is trickier than it was a few decades ago, since most of the motels and hotels around the region have folded. But you can rent cabins if you plan in advance from various outfits, and there are two quite nice national forest camp sites there plus oodles of commercial/independent ones. Seneca Rocks itself is a quite striking geological feature you can hike up and stand on top of, and Spruce Knob is just a hop, skip, and a jump away and is the highest point in WV with some great and very easily accessible views from the top. Don't forget to stop by Yocum's general store and pet the cats when you're there.

Dispersed camping is no longer allowed within the Seneca Rocks/Spruce Knob sphere of influence, but it is in the rest of the adjacent greater Monongahela National Forest, including in the Dolly Sods wilderness if you're into that sort of thing. Backpacking in Dolly Sods is quite possibly the best way to see the most varied terrain anywhere east of the Mississippi within the span of a weekend and without owning a private jet. The north, east, south, and west extremities of it may as well be on different continents; it's pretty wild.

Dispersed camping is allowed in Daniel Boone if you go there. You have to buy a permit to leave your car anywhere overnight to go backpacking but it's only a couple of bucks. The Red River Gorge area in Daniel Boone has some incredible sandstone formations including massive arches (some of which you can climb), shelter caves, cliffs, and overlooks. It's also home to the Nada Tunnel which is pretty cool but maybe not so appealing to people who are afraid of caves because it's basically a cave with a one lane road you can drive straight through. (It was actually originally a railway tunnel. I cannot possibly conceive of what it must have been like to cram a coal burning steam locomotive through that tiny passage, and if you see it you'll know why. But that's what they did back in the day.)

Civilized accommodations are easier to come by there including plenty of cabins and motels, and also hotels you can find near the interstate. If you're into rock climbing there are also a ton of climbing routes all over the Red River Gorge.

Forget Yellowstone. Yellowstone is so popular and yet so fragile and so dangerous that the entire place is on lockdown. You spend your entire stay there on rails, pretty much literally. Everything is boardwalks and pavement and everything else that isn't is cordoned off. Yes, this is so dumbasses cannot fall into geothermal features and be boiled alive. But it also has the net effect of causing you to take the exact same route in the exact same way and take the exact same pictures that everyone else already has. So you can have the same experience by just finding some rando's Flickr album or whatever and looking at their pictures, because they'll be just the same as yours. Plus the whole place stinks. Sulfur, don't you know.

Oh, and you get to contend with access roads clogged by all the dimwits from the midwest who stop dead in the middle of everything to try to fit baby bison into their minivans, or whatever the fuck else. I went once and that was enough. I came, I saw, I bought a mug. I have no desire to go back. (Nearby Shoshone National Forest, however, is friggin' awesome. So is Big Horn. Be sure to check out Shell Falls while you're there and annoy your nearest creationist.)

I would suggest Shenandoah National Park. Partly for the beauty of the park itself, and partly because the drive from Indiana will go through some gorgeous terrain. I would suggest taking route 33 through Wayne National forest in Ohio. You can stop in Athens as a halfway point, it's a picturesque little college town nestled in the hills.

Note on Yellowstone, Grand Teton is just south of it, and is much smaller, but significantly more dramatic (If you expect to summit anything there without a ski lift, good luck).

As for the east coast, New River Gorge WV is very active with guided activities, last I checked. One of the good places for rafting, also the Greenbank National Radio Observatory is within a days drive. If the Virginia(s) is your direction, in addition to the national parks, you also have the estates of some of americas founding fathers in the area, along with old battlefields that are open to the public, if history is on the menu. I would argue that the best park for hiking on the US east coast is White Mtn. State Forest in NH. Middle of ski country for the winter and summer is peak hiking for the best section of the AT (the Presidentials will kick you and your cars ass, Mt. Washington is up there and you can drive to the weather facility at the top)

Isle royal isn’t very easy to visit. But the parks department operates 2 “national lakeshores” in MI that everyone should see. Sleeping bear dunes is west of traverse city and is super accessible and close to lots of lodging in TC. Pictured rocks is perhaps my favorite place on earth. But, driving up to the UP is a bit of a trek.

I grew up going to the Great Smoky Mountains fairly regularly as a kid, my favorite was always to go to Chattanooga. It’s like a mid sized town in the middle of the Smokies, so it’s super easy to drive out into very rural natural areas during the day and drive back to the amenities of town at night. There’s also a lot of fascinating Civil War history in the area if anyone is a history buff.

The aquarium is great, and I’m a sucker for the incline railway and Rock City (featured prominently in American Gods). Rock City is technically in Georgia, but it’s a 15 minute drive away from Chattanooga so it might as well be right there.

If any of the kids are 8 or under they’d probably enjoy the Creative Discovery Museum. It’s a pity caves aren’t an option, because Ruby Falls is an underground waterfall in the area and very beautiful. The other thing we always did was either go horse riding or zip lining.

Oh, and if you see people selling fruit or veg on the side of the road with hand written signs, stop and buy it. It will be some of the best you’ve had. Same rule for boiled peanuts and apple butter.

The boundary waters area is really great, but maybe not for a family. Honestly, I don't think any of these areas are going to compare to Yellowstone in terms of the purely sublime combined with ease of access with children.

New River Gorge is nice, but, like a lot of these options, it's "take your kids for a weekend" kind of nice, not "my mom is paying for a multi-generational meeting at a national park visit" kind of nice.

It's all what you make if it though. Were I you, I would figure out how to go to Yellowstone. Especially considering it's the preference of the person, you know, paying for it.

I've been to all of those except congaree and if I had to go back to one I'd do isle Royale. Great for introductory backpacking with it's low elevation change and pit toilets at all camping areas. More modern toilets at either end of the island. You also can kayak to one of the the surrounding islands and camp there. If that's not your thing there's a hotel, restaurant, and boat tours of all the shipwrecks. I'm a big backpacker though so if I went back it'd be kayaking and camping on the chain if Islands around one end of the island. Lots of more family friendly options near rock harbor too. Like others have said you can visit lots of things in the way up to break up the trip. I'd do a day at pictured rocks, tahquamenon falls, kitch-iti-kipi, or even a couple nights on mackinac Island on the way up.

Shenandoah is really nice. None of the East Coast parks are going to be as mind-blowing as the famous ones out west. And you can also go to the Great Smokey Mountains area if you plan the route correctly.

Gateway Arch? I think you meant to say City Museum.

I think the Smokeys are the #1 most visited park

My partner and I have a goal to visit all the national parks, so we've done almost all of these recently.

Mammoth cave is by far my favorite, but I'm a computer nerd, and a giant hole in the ground is my natural environment, so there may be some bias.

Gateway arch is okay, but not worth a long drive in my opinion.

Got springs was very neat. It has a different feel than the other parks I've been to. It is a bit "touristy".

New River Gorge is amazing. Be aware that any hiking there is going to take a lot of vertical travel, a 1 mile trail there is more like a 3 mile trail in a flat area. That being said, it has a lot of wildlife, picturesque views, unique structures (big ass bridge) a cozy little town (Fayetteville reminds me a lot of downtown Bloomington) and plenty of activities (hiking, rafting, zipline, tree top obstacle course, history, shopping). Easily a top recommendation.

Congaree. Probably the most wildlife I've seen at a park. Snakes, salamanders, pigs, owls, fish, skinks and a shit ton of fuzzy caterpillars. Very buggy, but honestly the mosquitoes weren't as bad as everyone said they would be. Also highly recommend. The park has a long boardwalk as it's "main" attraction, which is very accessible. Saw lots of wheelchairs while I was there. I'd recommend taking a kayak tour as well.

Great Smoky Mountains is a great choice as well. Has that authentic "National park" feel. Very big (for an East of the Miss. park) Gatlinburg is nearby for lodging and other activities. You could easily spend a week here between the two. I like more unique "unique experiences" but GSM would be an awesome choice if your family isn't very adventurous.

Don't forget some state parks while driving through. Red River Gorge in KY is one of my favs, and easily worth the trip on its own. You've got Natural Bridge State Park right there, and the skylift is the best way to get to the top. Shout out to Daniel Boone coffee shop, best breakfast burrito I've ever had, but the line goes out the door, so be prepared to wait a little. FYI, this area had a bad fire last year, not sure about the extent of the damage.

All in, I'd say do GSM. Solid pick, easy to get lodging, lots to do if the weather doesn't cooperate. On your way over, hit up Red River Gorge on the way if it's not too far out.

Niagara Falls isn't too far out of your area, 9 hours for the American side, maybe 10 for Canadian. Walking on the pier under the Canadian side is incredible, if you are willing to pay for the ferry ride.

I've done quite a bit of camping in the UP of Michigan and have been really impressed by how much beauty can be found up there. Doing Pictured Rocks, Tahquamenon Falls, and Kitch-iti-kipi on a 4 day weekend trip is pretty incredible. So I'd do Isle Royale and hot up a few more landmarks while being up there. Sounds like a good clean inexpensive trip!

I live in VA and I find Shenandoah to be the most boring national park I've visited. Unless you can catch it during foliage season, I'd go somewhere else. That said, Skyline Drive is a mess during foliage season. Great Smoky Mountains is nice.

Having been to all but voyageurs go to either the smokey mountains or new river gorge. I was going to put mammoth cave on that list but, you know fears and stuff. I will say mammoth cave does not feel like your typical cave, it’s way larger, and has been adapted for tourists.

Not national parks, but Elephant Rock and Johnson Shut-Ins in Missouri are both pretty neat.

based on name: isle Royale

I've only been to Indiana Dunes and Voyageurs, but they're both nice enough. Voyageurs is like, bring a canoe and camping supplies, though, not necessary a day trip.

Consider some state parks! Turkey Run in Indiana is gorgeous, and Pipestone State Park in Minnesota is beautiful, historical, and unique.