I would agree that if you are planning a Spring trip, start South. I also agree that 3 weeks is not nearly long enough to do a loop including Arizona, Wyoming and South Dakota and really see much of anything other than out the windshield.
From Newmanstown you could take I-81 to I-70 to I-68 to I-79 to I-64 to Saint Louis. I prefer this route to I-70. In my opinion it is a more pleasant drive with less traffic and better road conditions. You could also take I-81 South to I-64.
You could take I-81 South to I-40 which has lots of potential stops along the way. East of the Mississippi there is the Knoxville, Sevierville, Pigion Forge, Gatlinburg area; Nashville; and Memphis. Continuing on West, Amarillo is good for an overnight with the
Big Texan.
Palo Duro Canyon is not too far to the South.
In New Mexico, there are a lot of things to see along the I-40 coridor including:
Tucumcari,
The Blue Hole,
Mining Museum,
El Malpais National Monument,
El Moro National Monument,
Ice Cave and Bandera Volcano,
Acoma Sky City,
Pueblo Cultural Center, and
Petroglyph National Monument.
Continuing West into Arizona, there is
Meteor Crater and
Petrified Forest National Park. You can stand on the corner in
Winslow, AZ. In the Flagstaff area are
Sunset Crater National Monument,
Wupatki National Monument, and
Walnut Canyon National Monument. South of Flagstaff in the Sedona/Cottonwood/Camp Verde Area are
Montezuma Castle and Montezuma Well National Monuments,
Tuzigoot National Monument,
Palatki and Honaki Ruins,
V-Bar-V Heritage Site,
Verde Canyon Railroad,
Fort Verde State Park,
Cathedral Rock,
Gold King Mine, and
Jerome State Historic Park. The South Rim area of
Grand Canyon National Park is not far North of I-40 and Williams, AZ.
With only 3 weeks, you will have to pick and choose where to go and what to see.
Hope you have a Great Trip whatever route you take!!