I think America should do it in baby steps. Eat away at gun laws.
Example.
1) How is it in America you can't buy a beer until you are 21, but you can buy a semi-automatic machine gun? That makes no sense. I'd up the age of gun ownership to 21. This would make it way harder for kids to shoot other kids. You can't be expelled from a college and go buy a gun. You aren't old enough.
2) I would make it an offense to carry a gun that isn't registered to you. You can't fire it, you can't hold it for a buddy, you can't look after it. The only person allowed to hold your gun is you.
3) You have to have your gun license on you, if you have the gun on you. Not carrying the license whilst armed should be an offense.
4) You have to obey drink drive style gun laws. If you are packing, you can be breathalyzed. If you are over the drink drive limit, you're over the gun carrying limit. The sentencing should be similar.
5) Ammo and guns must be stored in separate locked storage containers.
6) Designated smoking area style laws. You can't carry a gun into a school, into a hospital, into a restaurant, a convenience store, a bank or a bar. Anywhere where there are either vulnerable people, cash registers or they serve alcohol. If private businesses want to ban guns on their premises, its their private property and they are at liberty to do so and to search you before entry. You can take your business elsewhere if you object.
7) You can't operate heavy machinery when using certain drugs. That extends to carrying guns under the same influences. If you are too cognitively impaired to work a machine without hurting yourself, you're too impaired to make calls on when to pull out your gun.
Maybe some of these things are already laws, but I'd just keep turning the screw. Make it harder to get guns, more inconvenient to own them and pain to carry them around. If you really are a decent citizen and shoot targets on your own land or hunt and never bother anyone, I don't see an issue with that, but you wouldn't fall foul of any of the suggestions I put forward. You'd already be doing these things as they are responsible actions.