I agree wholeheartedly with the advice you've been given so far. New Orleans simply doesn't fit within your time constraints, especially with a large detour on one of the legs to visit a friend. It would only leave you a single full day in New Orleans and no time to see anything on the road. But, some other destinations you might want to look at for your RoadTrip adventure might include:
Yellowstone National Park: Two days there via I-94 with just
short stops, two days there, three days back
via I-90 with a day spent between the Little Bighorn, Devils Tower, Mount Rushmore and Badlands National Park
Rocky Mountain National Park: Two days there
via I-35/I-80/I-76 (note that I-80 across Nebraska follows the route of the old Oregon Trail along the Platte River), two days there, three days back heading first up to the Rapid City area (see above) and then home.
Great Lakes Loop: Up to DDuluth and across the southern shore of Lake Superior, Apostle Islands National Lakeshore, Pictured Rocks National Lakeshore, Colonial Michilimackinac State Historical Park, south along the eastern shore of Lake Michigan, Sleeping Bear Dunes National Lakeshore, down through Michigan to Sandusky (Cedar Point amusement park) and the Bass Islands, then home through Chicago and maybe the Circus World Museum in Baraboo WI. Four days driving, three days for major stops.
Great Smoky Mountains National Park: Two plus days there
via Chicago, two days in the mountains, and two days back
via St. Louis. That leaves you a day to spend in
either Chicago or St. Louis,
AZBuck