have a look at -
http://www.aopa.org/tsa_rule/ - which explains the hoops you have to jump through. You will need fingerprints taken - by the sherrif if you are in the US. You need the necessary forms completed by your chosen flight school to enable you to complete the TSA online system. Don't know how that affects you changing your visa once there but it allows you to apply for one if outside the US - and is killing off alien flight training in my view.
There was a full article on pprune posted by naples air centre you might still find it.
Oh and the savings in landing fees will more than cover the equipment you need. You might find it difficult to get the UK books in the US if you do the CAA syllabus. I currently fly in the UK on my FAA licence which although it is "valid" as an icao compliant licence does cause some complications with less knowledgeable rental organisations and particularly currently getting permission from teh French authorities to fly G-reg across the FIR to France.