I would split the trip, unless you have 2-3 months to do it all.
Fly to the East Coast, tour New England down to Florida and hop a plane and fly to the West Coast.
Otherwise, you'd only see road, but I'm guessing if you spend that much time and money, you want to get a memorable vacation out of it...
How much of Indiana, Kentucky, Alabama and Carolina do you want to see? Driving through farmland can get somewhat boring after a few days...
For an unforgettable trip, take a look at mapquest and streamline your destinations. Arizona & Utah (the canyons) are easier to reach from the west coast, Vegas would then be a given.
Or, if you really do have the time:
start in Miami, drive down to the keys (that's the picture-book Florida) for a day or two and then make your way up the cost.
That would give you:
Atlanta Georgia, Washington DC, Philadelphia Pennsylvania, New York and Boston Massachusetts.
In Boston, hop on I-90 and go west:
Albany, Syracuse, Rochester and Buffalo NY (including Niagara Falls), and Chicago Illinois.
Then all the central states: how much may depend on your patience, it does tend to get boring after a few days on the road out there. If you're flexible, decide while on the go.
Then aim for the canyons: Arizona and Utah, stop in Colorado too, it's worth it!
And on the way to San Francisco California, go through Vegas. Hotels and food are cheap there, so plan for a few days, before you continue to expensive California...
PS: If Americans do that kind of trip, they usually buy a car on one coast and sell it on the other. ;-)