Hitch hike? I don't think people actually do that in America. Take a bus or something. Also, the middle 2/3rds of America is pretty boring and is not going to be worth seeing. You'd be better off picking a couple of cities and staying in hostels.
I don't know if you've been to America much, but the country is very, very large. It's larger than you think if you grew up in the UK or Europe (or even a somewhat self-contained area of the US like New England). I don't even think I had a personal sense of how large the country was until I left Connecticut for college.
New York to Chicago to Las Vegas would be good. How about Boston to New York to Chicago to the South (Memphis?) to some place in Texas to Las Vegas to Los Angeles or San Diego. That trip would give you a good sense of the country.
You want to avoid long trips through South Dakota and western West Virginia and stuff like that. The vast majority of this country is large, flat land filled with good, simple people who love Jesus. Nothing wrong with that, but I don't know if that's what you're looking for.
Also - the NBA season will be over by the time you get here and football won't have started until you've left. However, if you hit Boston (Fenway) and Chicago (Wrigley), you'll be able to catch baseball in two of the greatest places to watch any sport anywhere in the world.
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