I don’t hear about people walking across the country very often, at least walking across America. But I’ve gotten super interested in long distance biking and that is very common for people to do cross-country trips. I’d suggest looking up some of that just to get more accounts of cross country travel. You will find a lot of the trails and roads that are legal to do these kinds of trips on.
I don’t think you’re likely to be thrown in jail at all. Be stopped by the cops so they can check you out, sure, that would happen quite a bit. But once they know you’re not intent on hitchhiking (I’m not sure if this is actually illegal, but maybe just a little dangerous), you’re not carrying drugs, you’re not insane, you’re not an illegal alien, etc, they’ll just wish you well and let you go on your way.
Honestly, after all the videos and stories I’ve read. I’d suggest doing this on a shorter distance to start with. I want to do a 100–200 mile bike ride by the end of the summer. But I’m going to be starting with 20–60 miles round trip rides to close cities, and out to Lake Michigan. With other people on some of them. Training and preparing mentally and physically for the long distance bike tour.
I don’t know if you’re ever heard of this/saw it, but there is this great documentary called Long Way Round. Ewan McGregor & his buddy Charley Boorman ride motorcycles around the world. From London to New York, over about 4 months. Its SO good, and it was one of the first things that spurred my interest in extended travel. The parts when they’re in Kazakhstan, Mongolia, and Eastern Russia are particularly inspiring. You can watch it all on Youtube here. They have another one where they ride through Africa, but I haven’t watched that one yet.