Create an additional narrow restriction to gun ownership for individuals with mental health problems, as a subset to an improved mental healthcare system for the entirety of country (which would also be a subset of trying to improve the giant clusterfuck of a twisted market that is general healthcare). This mental health problem list would require carefully maintained in regards to patient transparency as a component of medical records (as well as the required privacy requirements, which will be a fun balancing act) to make it clear to affected individuals when they are placed on it and the steps that have to be taken off of said list.
I would also be willing to provide a cursory background check to private transfers of firearms, provided that there is a clear system to ensue that information is not saved. I think that a phone call based system would be appropriate for it.
As long as I'm listing things, I'd do a serious gutting and rewrite of ATF regulations, but coming up with a good proposal for that would take me at least a fair bit of research and drafting work, so I'll leave it a hypothetical. At the least, I would greatly extend the ability of people to get cool weaponry without gunsmithing licences, probably combined with a registry system for the very energetic things. Destructive devices, collector weapon qualifications, and any other weapon would all get significant retooling as well.
I'd strongly consider doing something about state and local ordinances that are effectively "shall not issue" but that's really more a matter for the Judicial branch to strike down as they go, as part of continued incorporation tendencies.
Edit: fixed some instances of wording poorly