In no particular order:
Acadia - Smaller, but absolutely gorgeous, with a lot of neat history, too. Touristy in the summer.
Yellowstone - Interesting and pretty, but very touristy.
Grand Canyon - Beautiful. South Rim is very crowded. North Rim is more remote and thus less crowded.
Arches / Canyonlands - Incredible.
Everglades - Meh. Definitely cool to see if you're in South Florida, but, for me, I won't be killing myself to get back. I like mountains.
Tetons - Stunningly beautful.
Big Bend - Hot. Very pretty, but also very remote.
Carlsbad Caverns - Interesting, but a small park that's "do-able" in a day, unlike some of the others. Kids would love it.
Shenandoah - Very scenic. Close to other cool destinations (DC, Charlottesville)
Parks I haven't been to, but hope to shortly - Smoky Mountains, Rocky Mountains, North Cascades, Mount Rainier, Yosemite, Zion, Voyageurs, Black Canyon of the Gunnison, Cuyahoga Valley (because it's near my sister's house).
My family loves national parks. Pro-tip: if you're 62 or older, you can get a $10, lifetime pass to all of them. This includes admission for up to three others in your vehicle. Also can give discounts to other amenities, like campsites, etc.