Keep the Route Short and the Opportunities Plentiful
You'll need three full days of solid driving to make the trip. That really only leaves you two days for the sights and experiences you'd like to have. Since one of the mottos of this organization is that 'There are no boring places.' my first suggestion is that you keep the driving to a minimum so as to save as much time as possible for other activities.
So, the most direct route would simply be I-10/I-15 up through Las Vegas and into southern Utah, then I-70 east to Denver, I-76 northeast to I-80 just after crossing into Nebraska and I-80 the rest to the way to Omaha. That will leave the most time possible for exploring the sights near that route. And what sights they are.
Just to name a few possibilities... Besides cities such as Las Vegas and Denver, you'll pass near any number of National Parks (NP) and Monuments (NM) including, but not limited to, Mojave National Preserve, Red Cliffs National Conservation Area, Zion NP, Arches NP, Colorado NM, Breckenridge, and the Oregon Trail.
AZBuck