Eddy,
I did an intensive three-week PPL in Florida and I loved it. Weather is more reliable in the US than in Europe (with the exception of hurricane season) so it's largely your own ability which determines whether you can do it in three weeks.
Did you do a search on this forum, and the professional studies forum? A lot has been said on the various CAA approved (so you're issued a JAA PPL) schools in the US. I wrote a lot, there's a fellow SoCal Approach who made a fair number of comments as well.
Expect somewhere around 8500 USD for the whole course, including flying, exams, accomodation, but excluding the flight. So 10000 UKP should be plenty. But do budget some funds for a private groundschool lesson (about two hours) and four hours worth of flying lessons, for when you get back. Aerodynamics in Europe is obviously the same, but airspace division, R/T, weather, notams, legal, flightplans: there are differences and not all of them are obvious from the theory books.
Here's a shortlist of things you need to do:
- Pick a school. There's three in Florida (OFT, OBA and Naples I believe) and one in Southern California (AAA) that seem to attract a lot of Europeans. The full list of CAA approved foreign flight schools is on the CAA website. None of us here have been to all four of us so we cannot make comparisons. Read the reviews of the various schools on the forum (with a grain of salt: people are more likely to complain than to praise), confirm availability and do a preliminary booking.
- Arrange M-1 and TSA clearance
- Learn as much theory as you can beforehand (Trevor Thom or Jeremy Pratt books). When over there, you do not have time to study a lot.
- Try to get hold of an airband scanner and listen to R/T as much as you can, to develop an "ear" for this type of communication.
When over there:
- Try to pass all seven ground exams in the first week.
- Have a schedule prepared with your instructor, measure your progress based on this schedule.
- Remain polite, but be assertive. Be the master of your own education.
- Schedule and confirm your R/T practical exam and your skills test as far as possible beforehand.
Good luck!