Welcome to the RTA Forum!
The first thing you really need to know is that trips from NC to the Grand Canyon really can't be measured in hours - they need to be measured in days. Even the western most point of NC is a solid 3 day drive to the Grand Canyon, and if you are starting farther east like Charlotte, Winston-Salem or beyond you'd need 4 days each way. You already know first hand the dangers of driving drowsy, and trying to push beyond that would be a very good way to repeat your past mistakes.
Also keep in mind that even when you are talking about going 10-15 hours away, that time estimates that google and mapquest provide aren't realistic over long distances because they don't account for the basic stops required for every trip (food, fuel, restrooms, etc).
A good day on the road with few stops is about 600 miles, if you want to explore along the way 500 miles is a good daily max.
Otherwise, I think your instinct of not starting by trying to do a cross country trip is a good one. Starting by doing a trip a day or two from home will give you a good feel for what you like while on the road.
Then its just a matter of figuring out what strikes your interest. A day's drive away could include the beaches of Northern Florida (Atlantic or Gulf Coast), Land Between the Lakes/Mammoth Cave in Kentucky, or the West Virginia or Pennsylvania Mountains. If you want to push into 2 days you could look at New England, the UP of Michigan, or if you want to replicate some of the culture you'll be missing while your friends are in Europe, you could look at Montreal or New Orleans.