We just get the annual pass for the family. It is so worth it if you plan to visit more than 2 national parks in a year, especially if you have more than 2 kids.
Different places charge differently. One of them might charge by person (adult, child, seniors prices, up to $25 per person). Another national park might charge by vehicle and it varies ($7 up to $20 or more). And it all depends on how great and popular the national park is.

We've done 2 straight years of the Southwest US (UT, CO and AZ) during spring break. It was awesome. But, you also have wonderful state parks-don't miss those. Often, those have are better run facilities and better maintained trails because I'm told they have better funding.
We are tempted to go back to the SW but seem to be over doing it.

Focusing on the nearish:

Great Smoky Mountains NP is great for spring break.

Tack Jamestown & Yorktown onto a trip to Williamsburg or Busch Gardens

Shenandoah makes a great weekend trip. Once a month through the summer they have amateur astronomy events at Big Meadows. It's "amateur" but they bring some serious telescopes! You'll want to secure lodging at the Big Meadows Lodge or Campground, though, and that can book up.

Cuyahoga Valley NP (one of the newest NPs) is near Cleveland. It's 5hrs+ away and is more of a weekend park.

In the long drive / road trip camp you have:

Mammoth Caves NP between Louisville and Nashville (both cities would also can be great for kids to visit) is about 10 hrs.

Indiana Dunes NP near Chicago (one of the newest NPs).

Acadia NP in Maine.

Utah and Arizona are wonderful and some of my favorite national parks.

PP, can you tell us your itinerary for Everglades? I feel sort of guilty we've never been there, when we've been to so many NPs that are so much further, but I never really find the right time to go.

When my oldest kid was in 4th grade, we went to the national parks in Utah. For the summer after my 2nd kid's 4th grade year, we're going to the national parks in Colorado. You can do both in one trip, but we prefer to spend more time at each park.

We’re making the most of our 4th grade pass but FYI it’s only $80 and a great deal (also works at Great Falls, which I think is $20 a car now). We’ve gotten one each year for the last 3 years.

Best national parks road trips include:
Utah 5 + Grand Canyon
Yosemite/Kings Canyon/Sequoia
SD (Badlands/Mt Rushmore/Wind Cave)/Yellowstone/Grand Tetons
Olympic/Mt Rainier

Other favorite NPs: Joshua Tree, Death Valley (not in summer, obviously), Acadia, Everglades, Glacier

Olympic National Park

Yellowstone.

Yosemite in October is beautiful, too. No waterfalls but it’s quiet, reflective, lovely temps, pretty fall leaves.

Just FYI, we did Yosemite in late August when the California kids were back in school and there was no line to get in. Waterfalls are skinny that time of year though. Tons of Europeans in the park at that time of year.

I’ve looked into this as well. We plan to do all these road trips at some point, and it’ll be a bonus if we can do more than one trip in the 4th grade year.

Grand Canyon/Page/Bryce/Zion
Yosemite/Sequoia/Kings Canyon
Yellowstone/Grand Tetons/Glacier

Time of year you’re going probably affects your destination. Check crowd calendars and weather. For example, I want to avoid the biggest crowds in Yosemite when traffic in and out of the Valley is a nightmare, so I want to go in April. We’ll get to see the waterfalls at their best because the snow is melting, but will have to sacrifice some trails that are still closed. And I definitely don’t want to hike Zion in the dead of summer, so I’m leaning toward the fall for that one.

Glacier

Rocky Mountain NP is also great.
But you can use it at several parks in Utah, and that’s also a great trip.
I’m addition to a big trip, you can use it at:
-Great Falls
-C & O Canal
-Fort McHenry
-Harper’s Ferry
-Assateague
-Shenandoah

Yellowstone and Yosemite are magic, but for different reasons.

Yellowstone can be a non-stop parade of animals. It’s truly amazing to keep a list of all the critters you see. The drawback is that no other park compares when it comes to The variety of wildlife.

Yosemite is awe-inspiring beauty. You’ll pinch yourself. Very few animals, but gorgeous mountains and waterfalls. It’s a dream.

With either of them, you could also get to some other parks if you are willing to do some driving.

If you want a bunch of parks together, Utah is the spot. You can get to several parks in a 7-14 day trip.