General Aviation Flying in the U.S is much more hassle free, and easier then in most if not all European countries, szczególnie Polska, assuming you're not flying out of the center of NYC.
When you go check the airplane out, you can hang out at the flight school and I'm sure people will be willing to help and answer any questions you might have. Either way, don't stress it, and don't go memorizing the every FAR out there.
Main things concerning you to look into will be dimensions, and types of airspace, as well as airspace entry requirements.. ATC stuff, such as possibly creating a flight plan, if you wish to have one... otherwise you can probably just get flight following if you want it, and how to obtain it.. practicing English ATC comms if you're not used to them. Becoming familiar with the FAA VFR sectional chart printed by Jeppesen and symbols on it.. if you're not already. Legal docs required to be on you to operate the a/c (license, medical, photo id, etc), as well as legal docs required to be on the a/c (airworthiness, poh, etc.)
Powodzenia.