Everyone has covered licensing here but as someone who has just returned from San Diego doing hour building on my JAA PPL, I thought I might just give you a heads up.
If you hold a JAA PPL, get the FAA paperwork done well in advance. You need verification to go from the CAA to the FAA (the CAA kindly charge you £40 for the privilege) then in San Diego, you go to the FSDO on Aero Drive.
I did 62.5 hours in 3 weeks at Montgomery Field in San Diego which is also on Aero Drive, flying with Golden Wings Flying Club. I don't know about A-A Aviation but I did email them months ago about hours building and they never replied.
I loved flying at Montgomery Field. It lies in Class B airspace very close to Lindbergh Field and Miramar, so there's plenty to contend with. But actually flying in Class B is relatively straightforward, SOCAL Approach maintain separation and have you under constant radar surveillance and it's nice to get a call something like "Cherokee 1747T, traffic to your 11'o clock is Boeing 757 climbing out of 3000 feet, report in sight"
I thoroughly recommend San Diego and especially Monthgomery Field if you want experience of Class B. I feel that if you want to train as an airline pilot, you need to fly in this kind of airspace and not look for class G wherever you can. I spent time flying out to Las Vegas, the Grand Canyon and Monument Valley, all at 11000 feet in a PA28. It's an experience you'll never forget but be wary of the heat. When I departed Vegas for the Canyon and Monument Valley back to Montgomery, that day constituted 9.5 hours flying, and it's amazing how exhausting you find it.
But definitely go San Diego, and personally, Montgomery Field - if you like a challenge! I'm going to upload some photos at some point of my trip.