When I lived in Chicago and traveled west, my preference was to stick to the freeways and get to the mountains as quick as possible. That said, I have taken my time at various places in between.
Neche ND is in the far northeast corner of the state. I've driven the Chicago to Grand Forks and places west a number of times. Generally the more interesting areas along this route are along the Mississippi River. A stop at the headwaters is worth it (Lake Itasca Minn). The Red River Valley (Fargo north to Canada) is not that interesting - here the river crosses a wide flat floodplain, devoted to crops like sugar beets and potatoes.
US 2 all the way across to Glacier NP is an ok drive - mostly flat with a fair share of 4 lane. Teddy Roosevelt NP in western ND is an interesting stop, as are the Black Hills if you cross SD.
Heading west from the Black Hills I like to cross the Big Horn Mtns (US16 or US14). From Cody you can continue straight west to Yellowstone, or go NW on 296 to the NE entrance of Yellowstone. A side trip on US212 over Beartooth Pass to Red Lodge is one of the most scenic highways in the area.
From Tennessee, if one direction took me north to ND, I'd be inclined to pass through Colorado in the other direction. For example cross through Rocky Mtn NP, and then west into Utah before turning north to Grand Tetons and Yellowstone.
You can 'drive' and sample almost any of these roads via Google's Streetview mode.