What Difference Does a Day Make?
I make it roughly 1000 miles from Shreveport to Sanborn, MN (where US-71 meets US-14) the perfect place to 'turn left'. You could make that in two days if you stuck to the Interstates for the most part. But it you took US-71 I strongly doubt that it would take you more than three. Note that US-71 is duplexed with I-540 through Ft. Smith and the small cities of northwestern Arkansas, and with I-29 north of Kansas City (and you can always use I-435 around KC) up to St.Joseph. So you'll have some relief in your two most congested areas. Other than that, US-71 looks to be mostly a two-lane highway through the forests of the Ozarks and the farms of the central plains, avoiding most urban areas almost pathologically. That would leave you 4 days for your exploration of the Badlands and to start to work your way south along the front ranges of the Rockies in Wyoming and Colorado, and 3 days for the return home. So, in short, I like the idea of "US-71 north to Minnesota and turn left".
AZBuck
Little House on the Prarie
Apparently, my Mom watched way too much of that show when she was pregnant with me. The show was based on the series of books by Laura Ingalls Wilder about growing up in the US in the 1800s and travelling by covered wagon to various parts of the country.