You can do a correspondence program, but you really need to do an apprenticeship after it, as hands-on experience counts for an awful lot. They never mention this on the ads about these schools. Believe me, I went this route over 30 years ago, but apprenticed under a gunsmith who knew his stuff.
Next, not all courses are the same, and some would teach you some mighty unsafe things, as I have looked over a few available now.
If I were to recommend two, the first would be Modern Gun School, who actually makes you do some bench work for a grade. Next, would be Penn Foster, who is the same as the old North American Gunsmithing School, in Scranton, PA.
The rest, sorry, but I wouldn't recommend them.
Now, if you have the time to go to a professional brick and mortar school, then you can look into Yavapai College, Piedmont Technical College, Piedmont Community College, Murray State College, and Trinidad State Junior College. I also know there's one in Pittsburgh, but I heard they taught that it was all right to color shotgun frames with a torch, instead of properly case hardening them, so I reserve my judgment on them.
If you go to a brick and mortar school, then you can probably find a job right out of school, but with the above correspondence courses, you'll need to have work experience with them.
If you start your own business, you'll soon find just how expensive this work is, and that nobody will want to pay you what you're worth. You will find out just how high business overhead is, and when you charge for it, customers will balk.
These TV shows on the Discovery channel are not showing the many customers who take their gun, and walk out, and its not all glory either.
My last piece of advice is to learn to run a lathe, and a milling machine. You would do better to have a 36" lathe, and a small mill to make parts and tooling with. If you want to do refinishing, I advise you to learn how to make the finish match the factories, or else, you wont be doing it very long.