i am over here in the US doing flight training and I am british.
Its worth contacting the school to ask if there is any assistance they can give with the various documentation and proceedures as some schools do offer help with it. Firstly you will normally need an M1 visa, depending on it you have a different visa already, some of which allow flight training and some don't.
There is the the TSA, transport security administration, process to go through, which can take a while to get clearance, including having to have finger prints processed and cleared. Some schools will take you to get this done, some will leave you to deal with it yourself.
The TSA work can be started online before you arrive in the USA and is worth getting underway as soon as you are decided upon your school.