Good, fair question. Begs another question: what constitutes an illegal handgun? Is it some guy living in the middle of nowhere 30 minutes from police assistance possessing a firearm for his family's protection, or some great white hunter type who plugs nature's wildlife for kicks?
Also a good question. After all, in the US of A all illegal firearms were at some point a legal firearm, it was simply the act of an individual that changed its status.
I have also on many occasions fired guns, and jolly good fun it is too. I'm far from opposed to private ownership of firearms for sport, or for hunting (I find sport hunting highly distasteful, but we can't write legislation based on what we find distasteful no matter how hard Conservatives try). So, here's my manifesto for fixing the gun problem.
1. Hunting weapons are permitted, with a basic license (a criminal background check and a firearm safety course required, but permission is granted by default with those requirements satisfied). Hunting weapons are classified as single shot or manual repeating long guns (rifles) and shotguns. Hunting weapons must be stored in the home in a locking cabinet, unloaded, with the ammunition stored in a separate cabinet with a different lock.
2. All other forms of currently legal firearms, including semi-automatic rifles and handguns, may no longer be kept in the home or carried on the person. They may be stored and used at a licensed and insured gun club with a federally inspected and governed range, where you will be welcome to visit, check out your firearms and make blasty holes in pieces of paper to your heart's content. No license is required to do this. Should an individual not wish to do this, they may surrender their firearms to the government in exchange for payment of market value (determined as the original purchase price of the firearm depreciated over a term as determined by legislation). Under no circumstances, ever, is a private individual permitted to remove a firearm from the club - they can be serviced on site, and should an individual wish their firearms to be available at a different club they can request that the club transport them. All firearm purchases must be made through the club, as must private sales.
3. Concealed carry licenses will be available on a highly controlled basis to people with a strict professional need e.g. law enforcement, security officers, bodyguards and so on. Licensees will be subject to stringent criminal and mental health checks, including interviews, on an ongoing basis, and the license shall be valid exclusively to times when the individual is fulfilling their professional duties. Home storage requirements will be the same as for hunting weapons, and subject to unannounced audits.
4. Open carry is no longer legal for anyone except cops and security guards, mainly because of this absolute bell end:
5. Anyone in possession of a firearm in breach of the above rules is considered to be a terrorist and subject to prosecution accordingly.
I realise that this is a fairly significant change in your society, but that's kinda your fault for letting it get this far. Every other developed country figured this excrement out a long time ago, and unless people think that dead children in schools are a price worth paying to indulge saviour fantasies you're going to have to rip this band-aid off.