I concur with those who recommend the Southern route.
Stop off at the pissant little public use airfields in states like Alabama, Mississippi, Louisiana & Texas. The welcome you'll get at the mom & pop FBOs will blow you away. Really lovely people. They'll usually offer you the use of an old jalopy. Usually a piece of junk, but it'll get you into 'town' and you just top off the tank and give it a wash as payment.
Once you get into New Mexico and Arizona the rural feel at such places is just enchanting.
Make the Grand Canyon a 'must see'.