There are a couple of routes. The first (and to me, the WORST) is independent study. There are any number of folks that will sell you their training courses, DVDs, etc.
The second- find a smith to apprentice under. This CAN be good, IF you have picked a smith that is not only a good smith, but a good teacher. THAT can be hit or miss.
The third is to enroll in a trade school or community college that teaches gunsmithing. There are a few of these that have a program rated highly by folks that have been thru them- and you also learn some of the other things you need to know about running a business.
My personal advice would be to learn as much as you can about the art of being a machinist (tool and die maker type) and then grab a good school, or a good smith to apprentice to. Companies like Gander Mtn are hungry for smiths, and once you have had some training, would be a good place to gain some experience.