I'm surprised no one has suggested Route 66, the classic Chicago to LA road trip.

We've done a great deal of driving in the western U.S. I would most likely take the I-80/I-70 route as it's marginally the shortest and I'd want to get through the first part as quickly as possible. As suggested up thread, I would do the first part (until getting through Nebraska) as quickly as possible, and then take some time through Colorado and Utah to enjoy the scenery.

I would not miss a chance to get off the interstate and go through Valley of the Gods in southeastern Utah (south of Moab), as well as Bryce and Zion in southwestern Utah.

From Las Vegas, if you've never driven through Death Valley, it is also a fascinating trip.

Of course, it depends on how much time you want to take. The shortest distance as per Google maps is 29 hours of driving.