I would suggest the following: (based on other countries with low gun deaths)
1: In order to purchase a firearm or ammunition a person must pass a national license process to include: background check, mental health check, fire arm instruction on use, etc. This process will be renewed every other year. Additional license for a national concealed firearms certification that must be renewed every other year.
2: Sale of firearms and ammo will require a database check to confirm the license is current and has no flags, so no more 7 day waiting period. For gun shows buyers with a current license only will be allowed in. The seller will register the license number for any sales. For private sales the seller will contact local authorities to confirm the buyers license.