Douglas, Wy to Atlanta

I25 to I80 is the obvious choice at the WY end. If you are a GA resident, you probably already have a good idea of how to approach at that end. It's after Kansas City that Google Maps gives various options (that differ little in time or distance).

No it's not Douglas Az it's Douglas Wyoming near Casper

Are you going to Atlanta GA from Douglas AZ?
if so, take I 25 to I 40 to Nashville TN and then drop southeast to Atlanta.
or take I 10 to I 20 to Birmingham, Alabama to Atlanta.

Don't drive through St. Louis. Missouri and Illinois choose to use different route numbers for the same Interstate. Take I-270 N to I-255 S to I-64 E. Then I-57 S to I-24 E. to Chattanooga, then I-75 S.
I got so lost, I found myself in downtown St. Louis on Sunday afternoon, next to Busch Stadium when the Cards were playing for the pennant. I was so lost, that gas station attendants couldn't get me out of town.