Realistically, how much of the US could I cover on a bike in three months?

i used to do first aid on bike across america folks. The most common injuries were carpal tunnel syndrome, ischial bursitis, tendinitis of knee and hip flexor issues. You have alot of time to learn injury prevention, find the most adequate bike and plan a good route. It is nice to have company, or to talk with those experienced to gain an insite on appropriate routes. Southerners are notoriously evil. They might chose to shoot you off your bike...see the movie Easy Rider. If its summer, Go North, young man. Better routes over all. Once tyou hit the W coast, North or South. But avoid the desert SW like arizona, texas and new mexico. Texas is way to dangeroous and they are not bike friendly. Expect those knuckleheads to try to run you over with a great big pick up. besides all they ahve to eat is road kill. Good LUCK!