Google "FAA piggy back license" and you'll get the idea.
The best bet though it choose a big ish FBO/school that you plan to rent from and ask them to set you up a piggyback licence - big places will be used to it and will talk you though the process, forms and usually have someone at the FAA field office who can smooth the way (it's a morning's activity on a good day).
You'll need - you valid UK licence, valid UK medical and that's pretty much it.
When you have the piggyback to activate it you'll need a flight review (what used to be called a BFR) and hour of ground and an hour in the air - the former you'll need to be comfortable with US airspace etc and the latter will double for your renter check out, so it's not a great imposition - whole thing a day if set up in advance.
The piggy back is valid for life so long as your UK licence is valid (in UK BFR and have medical) - you can sub a US BFR and medical for the UK one if the UK rules make it difficult to obtain one.
Also you can piggy back only once - so if you have SEP and Rotary and piggy back one then you can't piggy back the other one.