The following are things that people in the majority party of Congress and the executive branch have said they would support:
Moving semi-automatic rifles to 21 years old ownership in line with pistols (or at least assault weapons, both ideas have been floated)
Hundred percent background check
Improve background check system
Additional General funding to mental health services including more inpatient beds funded
Teachers being armed, potentially with funding for said training
I don't have an issue with any of these honestly, except the arming teachers thing. I don't mind a teacher having a CHL. I think the law should allow it. I wouldn't give incentive for most teachers to be in that position. I've done combat pistol training and frankly it's not easy. You have to drill it like you're a cop, shooting all the time, hundreds of rounds all the time, to maintain any level of adequate skill without being a danger to others.
I also have a concern that the background check system for which technological solutions exist right now...But I am suspect of the government bureaucratic ability to put together a quick and efficient system that doesn't hinder honest gun owners transacting legal purchases and changing firearm ownership.