Depending on your children's interests and your determination to drive the whole trip here's a sample itinerary. Pick up the local travel brochures along the way to see if anything is deemed visit worthy for your and children's tastes. Your avatar makes me think Space Museums may be of interest?
East bound: I-90
Devil's Tower (WY) (have kids watch Close Encounters sci-fi film)
Black Hills SD Mt Rushmore Rapid City
across I 90 to Lacrosse WI then go south following upper Mississippi to Quad Cities
East to Chicago (Hancock Tower observatory, Navy Pier,)
East to Sandusky (Cedar Point amusement park - ride roller coasters)
Cleveland/Akron (Rock Hall, Stan Hywet Hall) avoid CLE the weeks of July 10 through 21st) due to RNC
Pittsburgh approach via Washington tunnel so they can see the grandeur of a city appearing out of nowhere
visit(Carnegie Museums, ride funicular up Mt Washington)
if they like Baseball catch a game at either CLE or PGH both have great setting of ballparks in downtown.
Go North to Rochester (Letchworth State Park, then George Eastman Museum/ Mansion)
On to Chimney Bluffs / Sodus Point on Lake Ontario
then south along Seneca lake to Watkins Glen (hike the glen)
Watkins Glen to Cooperstown (Baseball Hall of Fame)
Cooperstown to Albany then north cross Lake Champlain on US Rt 2 then to Montpelier
Montpelier to Concord to Portland spend time on Maine Coast
unto Boston to Plymouth landing site
Then to either Springfiled (NBA Hall of Fame) or Hartford to Samuel Clemens home then south to Atlantic coastline to NYC (Ellis Island, whatever the kids like)
Philadelphia Liberty Bell / Independence Hall
Gettysburg blue ridge parkway south to Asheville (Vanderbilt Mansion)
Asheville to Charleston (Antebellum south)
Charleston to Savannah
Savannah to St Augustine (south to Cape Canaveral Space Museum) then onto
Sarasota (Ringling Brothers museum)
Tallahassee to Montgomery to Huntsville (another space museum)
North to Nashville and onto Mammoth cave and Bowling Green KY (Corvette factory / Museum)
Down via 68/79 (Land between the lakes)
S on 45 to Natchez Trace to Port Gibson (Union army Gen. refused to destroy) then Vicksburg (civil war battlefield memorial) Antebellum south architecture and onto New Orleans.
New Orleans (enjoy the cuisine, French quarter) to Austin TX (State Capitol building)
west unto Guadalupe Mtns, to (Carlsbad Caverns and Hueco tanks on way to El Paso.
El Paso to La Mesilla NM and Las Cruces (White Sands for sledding sand dunes)
I 25 north to Denver, Cheyenne, seas of Sunflower fields (Scottsbluff, Fort Laramie)
over to Tetons and Yellowstone (20 to US 86 to US 84 follow Snake River valley)
Boise, then Ontario OR, (Four Rivers Cultural center)
and return back to Seattle
Whew that was fun... you're a great mom!
As someone else mentioned, when you do road trips, I found getting to your destination for next sight to be best done late afternoon and evening hours. Then get up early, stay at places that have complimentary breakfasts (the low end motels run the gamut so some strategic grocery shopping will be needed) and then go directly to the visitor / tourist site first thing and spend however much time you need. Many of these I have been to too but in much shorter excursions than you are planning. A few are still on my list.
Good luck and Safe travels