Your first step is to ask yourself a very important question - which country do I wish to issue my JAA licence. Having answered that then you need to consider each countries own interpretations of the JAA FCL rules and regulations. For example in the UK the CAA have a list of approved flight training organisations (FTO's). This is available on their website and should you wish to go down the UK licence issue route then you need to acquaint yourself with that list. Then you need to check if your US/SA schools are on it. Remember also that the list does change as schools gain/lose approvals so it isn't cast in stone.
You say that you might want to travel to the US to save some cash, well that is fine. Now assuming you are going down the modular route (as opposed to integrated) then as was pointed out above there is no 1 school that will give you everything you need. The CAA have a rule that the IR skill test must be conducted in JAA land ie: NOT the US.
To fly professionally you will need the following and this is the usual order that it is done in:-
1) PPL + night rating
2) ATPL ground school
3) Multi rating, CPL and IR
4) MCC
Now 1, 3 and 4 can sometimes be done at one school depending on what each one has approval for. You have various self study routes for the groundschool (called distance learning), plus inhouse residential versions as well. Schools such as Bristol, BCFT,Oxford, Cabair etc etc will all sell you a set of 14 manuals and let you get on with it. When you are ready you attend a brush up course of usually 2 weeks duration at their base and then travel to an exam centre to take the papers. In the US I am not aware of anywhere that you can do 4. Naples Air Center used to have approval to do 1,2 and 3 but has lost its JAA flying approvals for the moment.
Also there are various rules and regulations concerning the hours that you must have before moving on from step 1 above. For people contemplating going with the CAA then these are contained in a document called LASORS which is again on their website and has been linked numerous times on pprune so a search will find you that. It will also help you answer loads of other questions such as the format of the ATPL exams, number of sittings you are allowed, how long you have to pass them etc etc. All good reading for when you are losing the will to live.
Note that I haven't included hour building on my list. This is usually done after step 1. Some people keep flying all the way through the ATPL ground school and keep their hand in. Others feel that it distracts them from study. Clearly where you do it and how you go about it will depend cost and what suits your circumstances. Many of us who post here on pprune have saved some cash and gone to the US. You will probably want between 50-100 hours so the costs are significant. Some people buy shares in aircraft, some rent from a school. There are many ways to skin a cat.
There is also a sticky at the top of this forum which you need to read and get a full understanding of. To answer your question on cost, well there is no cheap place. If you find it please let us all know.
As a rough estimate I would expect that if you pass everything in minimum time and found the cheapest schools for each bit of the course then you won't have much change from £40,000. That excludes the living expenses, accomodation, flights to the US/SA, licence fees, ATPL exam fees, buying flying equipment (headsets, kneeboards, calculator, bags, pens/rulers, CRP5, torch, stop watch etc etc), beer money, landing fees and just about anything else you want to throw in.