Mine Gott, you are not even supposed to even fly a light aircraft in the US without at least a student (M1 for many courses) visa AND TSA approval. Mine a valid for 5 years, but I also have R stamped on the visa (A do what you like).
Mind you, I've flown all sorts in the US and no one even asked me for my passport, but do at least try and get the TSA approval, as you can run into real trouble trying to use a larger airport, an international one in particular.
The pain with an initial TSA approval is that manual finger prints are required. After that it's no big deal and just an online job. Any student visa requires an official interview at an embassy or consular office.
The FAA might check your immigration and TSA status when issuing a new rating or license, as will the Police if you bend a hire car.
You need a school or company sponsor to get a student (Or work) visa, or a TSA approval and it will take time if you have an interesting passport or family background.
NEVER PAY ANY SO CALLED VISA OR TSA GROUP AN UP FRONT FEE, as there are some real cool scam companies for work permits in particular. Only pay the correct online embassy or TSA web site fees (Plus the fingerprint lady in cash, which should be around 100 usd).
IF A COMPANY OFFERS YOU A JOB, LET THEM PAY FOR THE VISA, if they ask for anything up front, like a type rating deposit, the job does not exist even if the company does.
A LOT OF FAR EASTERN OFFICIALS LOST THEIR SHIRTS LAST NIGHT AND THEY WILL TURN TO A WHOLE BUNCH OF SCAM GAMES, including false airline jobs, to make money. Because those fraudsters are already active, their offers will be difficult to spot, so always ask for one months pay and a free return ticket if you already have a license. 