Niall It is up to you but there are a few things that you should consider.
First and foremost, Do you have the right to live and work in the USA? (green card or US passport)
If you don't and you can't get one then flying for a US airline is not going to happen.
If you do then keep in mind that in order to get a job with a major airline over here you will need somewhere in the order of 4000-5000 hours and thats not just any kind of flying, ideally the majors like to see PIC in airline operations, so the typical succesful profile is someone who has been instructing or delivering cancelled checks for two years or so and then flying for a regional airline for a further 3 or 4 years with at least one year as a regional captain.
This is what you will have to do unless you are lucky enough to get into the military, in which case your commitment will be for ten years.
The salaries only get good when you get to the majors so you are looking at a significant number of years on the breadline. Instructiing pays poorly My CFII with 3 years instructing experience is taking home about $1200 a month (his rent is $600) and he is working at one of the best and busiest flying schools in California.
If you are prepared to put in the effort then the rewards will not dissapoint, flying for a major according to the dozen or so pilots that I know who do so is an unbeatable experience.
One other thing to consider is that FAA licences and ratings (with the exception of the flight instructor rating) never expire so even if you decide to go JAA it may pay off in the long run to do the FAA first and then convert, you never know what could happen down the line.
As far as FAA training is concerned there are a number of excellent schools over here, a large group of good schools and a significant number of poor schools, however you usually get what you pay for and if you stick with part 141 schools you can usually be assured of good quality, the signs to look for are how long they have been established and if the offer a speciality (such as aerobatics ) these outfits tend to attract better instructors especially retired airline and military pilots who work part time, however their experience filters through the whole organisation, I've seen this with two schools, one in LA and the other in the Bay area.
Good luck with whatever you decide