Here's a few suggestions...
From Mount Katahdin, Maine, I would
cross over into Canada to Quebec City...it's a unique, beautiful city.
Head west to Montreal, check out vieux (old) Montreal area, very nice.
As other poster suggested, head south to Plattsburgh,NY.
Go west on rt 3, check out Saranac Lake and Lake Placid,
enjoy the amazing scenery of the Adirondack Mtns.
Continue west and check out 1000 Islands area (Boldt Castle, etc)
From 1000 Islands you can either cross into Canada and head west on HWY 401 to Toronto,
or head south on I-81 to Syracuse,NY then west on I-90 to Buffalo/Niagara.
Either way, definitely check out Niagara Falls area on both sides of the border.
Fort Niagara,NY Fort George,ON Niagara on the Lake,ON
Continue west either on I-90 (or via HWY 401 to Detroit then west on
I-94).
From Chicago continue west on I-90.
Just west of Kadoka, South Dakota, exit 131, turn off I-90
to check out the Badlands. Amazing drive, do not miss it!!
At Rapid City,SD head south into the Black Hills.
Another must see is Mount Rushmore.
Near Deadwood,SD turn back onto I-90 head west thru Wyoming.
As an option you can make detour north to see "Devil's Tower"
I did, it was OK, but you could skip it to save time.
Near Sheridan,WY head west on rt 14 to Yellowstone, a definite must see is "Old Faithful Geyser".
North on rt 191 to I-90 at Bozeman,MT , then north on I-15, west on rt 2 to Glacier.
IMO Glacier National Park is amazing, if you can, I would cross over to
Alberta at Chief Mountain then take the tour boat from Waterton Lakes
National Park into Glacier National Park, if possible, stay at the Prince of Wales Hotel (awesome views).
Head west thru Glacier on "Going to the Sun" hwy
....all I can say is wow, can't describe it, you just have to experience it.
From Glacier head south on hwy 93 to I-90 at Missoula,MT
Depending on if you want to check out Seattle or Portland (or not)
Either keep west on I-90 or south on hwy 93.
From Los Angeles I would stop by Las Vegas via I-15 then east to
St.George,Utah.
Like others mentioned, Zion is a must, from there head south into Arizona to Grand Canyon North Rim.
Next, north on rt 89, Page,AZ a good place to stay for the night.
From Page,AZ, east to Monument Valley, one the most awesome places I've ever seen, a definite must see
Continue east, check out "Four Corners" monument, and "Shiprock".
I personally recomend seeing Taos, Taos Pueblo, and Santa Fe.
Just past Santa Fe, cut down to I-40 via rt 285.
As mentioned before, don't miss eating at "Big Texan Steak Ranch" Amarillo,TX.
Just south of Amarillo is Palo Duro Canyon State Park, also worth checking out.