This is an issue that has been greatly debated in the classic aircraft community, not only by museums but aircraft operators as well. It is best to divide into two distinct sections, those that want to restore an aircraft to STATIC museum quality, and those that want to restore an aircraft to LIVE airworthy trim.

I will deal with the former first: There are organizations that run courses on the restoration of aircraft to museum quality and I believe they have already been mentioned several times in this thread. but quite frankly as long as you are enthusiastic, willing to commit your time (and occasionally money) and suffer the incessant nagging you will receive, when you have spent the entire weekend up to your elbows in the corroding mass that you are trying to restore to it’s former glory, instead of cutting the grass or taken the mother-in-law out to the local shopping village, then the only limiting factor is how authentic you want things to be.

And now the Latter: Here things become complicated. Unless you have proven experience of aircraft work (i.e. an apprenticeship) the CAA will not grant permission for you to work on aircraft by yourself. There are occasions where the CAA will grant permission for people with some engineering experience (i.e. car mechanics etc) to work on live or restoration to fly aircraft so long as their work is inspected and signed off by a Licensed Aircraft Engineer, preferably with a rating on that aircraft type, at every stage of work. If you are a time served apprentice you are able to work on an aircraft, until it becomes necessary to perform duplicate checks at which point a licensed Aircraft Engineer is required.
This section can also be divided in to several sections. These are, permit to fly aircraft (typically ex-military aircraft), Special C of A aircraft (aircraft capable of carrying passengers but not on a fare paying basis) and full C of A aircraft (aircraft that are either charter or part of an airline to carry passengers). This again brings varying degrees of operational requirements from the CAA and qualification requirements.
Over the years several companies have offered apprenticeship schemes, Air Atlantique, Jet Heritage and De Havilland Aviation to name a few. Of these only Air Atlantique has actively pursued the recruitment of candidates but unfortunately this has since ended. Some companies and collections have at one time or another offered cross training on other aircraft type i.e. if you had solely jet experience you could normally find an organization or individual who would be willing to train you in piston engines or fabric repairs.
Unfortunately now, the classic aircraft industry is in decline. The skills base that was around 10 years ago no longer exists and there are ever diminishing numbers of people interested in joining the community. Those that do find themselves in an industry that is woefully underpaid, where training is non-existent and where officialdom does everything in it’s power to get rid of them.
To all young people reading this I have this to say as terrible as it maybe. I am your age, I have been in this industry now for nearly 5 years. during that time I have gained valuable experience from some truly wonderful people and I have throughly enjoyed myself. BUT, I have incurred massive debts, the wage I am paid can’t support myself or my family and I am apparently in the top end of the pay scale for the industry. This is not a sector that offers great potential for career prospects unless you are able to provide the finances to gain your licenses. I am now seriously considering my position in the industry not only because of the pay but also future training, career prospects and the ever present specter that the next CAA regulation (in cahoots with the EU) will finish the industry.
The maintenance of classic aircraft is becoming the domain of the retired, soon to be retired and the rich. This is a state of affairs that will not be able to sustain it self for very much longer, as the remaining skilled people either move on to better paid jobs or move off this mortal coil. I personally bitterly resent having to move out of the industry for the reasons I have given, as I dearly love the work I do and the people I work with. There are very few industries that have the camaraderie and pleasure of coming to work that this does, but unless something changes the industry will die on it’s feet.
In short you are best starting your career in the commercial world with people like Marshalls, Serco, Fleetlands or an airline and moving to classic aircraft when you have a good 20 years experience under your belt.