Taking them in order,
Apprenticeship.
Depends on finding a willing teacher.
Second, finding one who's A. a GOOD gunsmith, and B. is a good teacher.
I've seen more than a few cases of hack "gunsmiths" teaching other people hack gunsmithing techniques.
With an apprentice setup, you're in the position of not knowing if your teacher himself is worth a damn and if what he's showing you is valid.
I've had people tell me that they had apprenticed under a "just great" gunsmith and they really knew their stuff.
Many of them I wouldn't trust to tighten a screw on a Daisy BB gun.
In many cases, it wasn't their fault. They just picked a shade tree gunsmith to learn from, and never knew the difference.
Military.
The military teach you to be a REPAIRMAN on military weapons. A good bit of this is to more on parts switching to get the gun in spec, than doing actual repairs.
You have to be careful about what you sign up for. Most military gunsmiths are really nothing more than good armorers.
Most employers are looking for MORE than just an armorer. They want a real gunsmith, although some companies that make AR-15, M1A, and 1911 pistols do hire straight armorers.
Most employers want people who also know commercial firearms.
Home study/correspondence.
You can learn this way, but most employer's will blow you off if you apply for a job with a certificate from a home study course.
The big problems with home study are in two areas:
First, since the instructor isn't standing there looking over your shoulder at your work, you never really know if your doing it the best, fastest way.
Next, without the hands-on instructor, you never know if your doing the RIGHT way.
The project may go OK, and it may work, but it's very possible you actually did it all wrong. Without the instructor, you never really know.
Last, there's one big service a hands-on teacher can provide: He can tell you that you just flat ain't any good at it.
This is something not talked about much, but in most professional schools, like law, engineering, or (thank God) medicine, the instructors can spot an unqualified person who just doesn't have the talent or skills to do it, and will in one way or another let him know.
I saw this in watchmaking school and in the gunsmithing business There were people who wanted to be watchmakers or gunsmiths who just plain didn't have the talent, the eye, the manual skills or whatever.
These people either figured it out on their own, or in extreme cases the instructors had to have a "Dutch Uncle" talk with them.
The better gunsmithing schools will do this. Correspondence schools can't, since they don't actually see your work.
Trade school:
If you want to be a PROFESSIONAL gunsmith go to a good school.
Colorado School of Trades, Trinidad College, Yavapai, and Lassen in California all are top professional schools that turn out REAL, professionals.
They turn out people who can go to work for industry, gun makers, as civilian employees of the military, and for local, state and federal law enforcement agencies.
Most of these schools have placement services, and they get visited by employers looking for good people.
So, if you want to learn how to do repairs to military weapons, join the military. If you're no good at it, they'll FIND a job for you somewhere else in the service.
Want to learn about working on guns that may or may not be right, apprenticeship, IF you can find one.
Want to be a hobbyist and play around with guns. Correspondence.
Want to be a fully-qualified hard-core professional, go to a good school.