Is training on a firearm a requirement to buy one?
Depends upon the state. The laws are very different depending upon where you live. And it depends upon what kind of firearm you are buying and what you are going to use it for.
For example, concealed carry in most states requires taking a handgun course, generally the one the NRA teaches and in Ct. if you want to buy a handgun you literally have to have a concealed carry license to do so. It is four hours of class in the morning and four hours at the range in the afternoon. That is how the course is given in Florida and Ct., my home state, has a similar requirement. Once the course requirement is satisfied you then go to a government office and apply for your state permit and pay the fee. In Florida that was a simple exercise taking less than a half hour. In Ct. the process requires getting a town permit first from the mayor or the police chief. The local police do the state and federal background check. Then you go to a state office to apply for a state-wide permit that is only good in Ct. The total Ct. process can take up to four months though a state law says it can take no longer than 8 weeks. Ct. has no reciprocal agreements with other states. Florida does with 37 other states. In NJ concealed carry is an "as needed" process. You have to justify why you want to carry concealed.
If you want to go hunting then you first have to pass a Hunter Safety Course for either gun hunting or bow hunting in many states especially in the northeast. Most states will honor a hunter safety course certificate from another state. This is usually four hours of course work in the morning and then a few more hours at the range shooting. The bow course I took required an accuracy component to insure you could hit what you were shooting at. If not, you failed. Interestingly, they announce this requirement again at the beginning of the morning session and usually about a quarter of the class gets up and leaves.
So the gun training is to hunt, not necessarily to buy a shotgun or rifle. To buy either in Ct. requires taking a gun safety course and then you petition the state to get a Long Gun Certificate which allows you to buy either a shotgun or rifle. The process to get the Long Gun Certificate requires picture ID, finger-printing and background checks. There is even another Certificate to buy ammo. So some states require training even to just buy a gun.
So, why doesn't a Ct. citizen just go across the border to another state with easier gun purchase laws? Because there is a Federal law that says that when a citizen buys a gun anywhere in the U.S. from a licensed gun dealer (one holding a Federal Firearms License, like Cabela's or your local gun shop) if that seller is in a different state from the state in which the buyer is domiciled, then that gun must be shipped by the selling FFL to a receiving FFL in the buyers home state. So if I buy any gun in Florida when I am there in the winter from an FFL shop they have to send it to an FFL in Ct. and I can get it when I return home in the spring. But I have to conform to Ct. laws and have the correct purchase Certificate, either a Concealed Carry license or a Long Gun Certificate depending upon what I bought in Florida. Generally, if you have a Concealed Carry License you can buy any kind of legal gun and ammo so in Ct. we have thousands of citizens with CCLs but don't even own a handgun and will never actually do concealed carry.
So this was an easy question which most people probably have thought about at some point but the answer is terribly complicated with 50 different answers depending upon what state you live in and where you are buying the gun. And we are only discussing purchases of guns from an FFL and have not even scratched the issue of private sales, which are also regulated in Ct. requiring a Ct. state resale certificate but Ct. is probably the only state that has this kind of control over the used gun/private sale market.
The bottom line is the vast majority of the U.S. population has very little knowledge about the existing gun laws in the U.S. The continuous call is for more "common sense" gun laws but many of those are in place already.