My standard technique would be to look at one of those maps that has a green dotted line on the scenic routes. Surely, there must be some better ideas.
Think about my predicament. I love to drive in the landscapes that begin about 700 miles west of where I live. I don't want to fly somewhere and then rent a car. I want my car. I'm tired of devoting Day 1 to getting past the boring by using the quickest route and occupying my mind with audiobooks. I want to make the driving itself interesting.
Take into account the need to find a place to stop for the night. In these vast expanses of America, it's hard to find a decent motel, let alone a place that I'd actually feel good about. Camping is not an option -- as you probably know by now.