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There are probably an uncountable number of ways to get from one corner of the country to another, so I'm going to give you just one that I would find reasonable in length, and possessing scenic, historic and maybe even quirky attributes. But your restriction that don't want to go "hours out of the way" is actually pretty significant given that the shortest possible route clocks in at around 45 hours of driving time, a minimum of six days.
Fortunately that route is fairly scenic and historic in its own right. It would be I-75 to Chattanooga, I-24 to I-57 in western Kentucky, I-57 to Mt. Vernon IL, I-64 to St Louis, I-70 to Kansas City, I-29 to Nebraska City, IA/KS-2 to Lincoln NE where you'd use US-77 to connect to I-80, I-80 to I-84 near Salt Lake City, I-84 to Portland OR and finally I-5 to Salem. Atlanta, St. Louis and Portland are the only large metropolitan areas you'd have to negotiate and all have beltways that would let you avoid driving through downtown.
A few stops, scenic areas or short detours along that route would include several Civil War battlefields between Atlanta and Nashville including Resaca, Chickamauga, and Murfreesboro/Stones River. You could then Use US-68/I-69 between Cadiz and Calvert City KY to explore Land Between the Lakes. St. Louis has several places worth visiting, my favorite being Grant's Farm, home of the Budweiser Clydesdales, but your next big stretch of interesting road would be I-80 through Nebraska which follows the Platte River and the old Oregon Trail. Just off your route north from Salt Lake city is Willard Bay State Park for a chance to experience a unique body of water. In southern Idaho I-84 follows the Snake River, but this is also a chance to get off the Interstate and follow the Thousand Springs Scenic Byway. Most of the rest of the trip would be through the Blue and Cascade mountain ranges and the scenery should take care of itself.
For smaller stops all along the major Interstates, check out these listings.
AZBuck