In order to help tackle problems regarding gun violence while still complying with the right to bear arms, I think it would be beneficial to regulate the usage and sale of firearms similarly to how we regulate who can operate a motor vehicle.
In order to legally own a firearm, make background checks mandatory, make gun safety classes mandatory, and issue licenses to legal gun owners proving that they've gone through the necessary steps to own a firearm. Perhaps every certain number of years, make the license be renewed as one would a drivers license.
Finally, close any loopholes regarding private firearm sales and fine or prosecute anyone who willingly sells a firearm to someone without a license. (Not unlike you would with a bartender serving those without ID.)