I have a suggestion that might not go over too well. Ride a bus. You'll get to see a lot of countryside, you won't have to worry about trespassing, and you'll get to where you think you might like to go much faster.
I think the idea of riding across the country is a great one...for experienced riders. I don't think it's something you should take up two or three months after graduating college.
Whatever you decide, I hope it all works out well for you. Oh, and if you would like a hybrid of the two, take a folding bike with you on the bus!