If you have a JAA PPL and Medical its pretty straight forward to get a an FAA PPL.
Firstly you need to down load the relevant forms from the CAA website ( sorry don't have the link to hand) you also need to down load the FAA forms for converting your license. Once you have these Fax them off to both the FAA( to the nearest office to where you plan to fly in the USA ) and CAA. The FAA will contact the CAA to get confirmaiton of your details etc. Now the FAA don't charge you but the CAA will! (suprised?!) You will then get confirmation from the FAA that all is in order.. allow about 6 weeks.
When you are in the USA you then have to go to the FAA office and have a brief interview. Upon which they issue you a temporary airmans certificate. You then have to carry out a Bi-annual ( or is ennual ) flight test with an instructor and then you're good to go. You shouldn't need a training visa unless you plan on doing specific training course.
There are a few posts on this subject in the Private flying section which may give you more information.