Maybe not the only thing, but certainly one major thing. There will always be crazy people out there, may as well attempt to keep guns away from them.
What else can be considered? More security for schools/theaters/malls? That will not make people feel safe, and a shooter could easily shoot the security officer. Having multiple security officers at every public place would cost millions in taxpayer money, not to mention: what kinds of countries have security rooming the streets? VERY scary ones. I would not want to get a pat-down and sent through a metal detector just to pick up some milk at the grocery store. That does not sound like a safe, free country. That sounds like prison.
Increasing background checks before handing out guns is gun control.
Maybe test everyone when they hit 18 to see if they are mentally stable? That would be ludicrous. What would you do if they aren't? I'm bipolar, would that be reason to throw me into a mental hospital, or in any other confined area away from the rest of society? Hell no. That sounds like segregation. Not allowing me to have guns? That would make sense, but again, that's a specific gun control issue.
Teach kids self-defense? Unless you're teaching it with guns, no child can defend themselves against guns, and guns in the hands of children? CRAZY TALK.
Harsher penalties for offenders? That won't help. Most of the time these kinds of shooters kill themselves or turn themselves in. They don't care about punishments. Murderers who commit single-person or serial style killings generally don't think they will be caught. They try and hide evidence, flee the country, etc. Stricter punishments won't make even 1% of murderers think twice.
I'm unsure of what else could be done that would have a positive impact on American society than gun control in any shape or form. It doesn't have to be a complete banning, but obviously gun laws in the United States are far too lax.
EDIT: Hopefully edited in time for others to view these links. Rightfully so, this has already sparked a gun control debate all over America:
Daniel Stewart @dnstewart67 tweeted, "Any American who would prefer to see this on their TV rather than lose the 2nd amendment have no humanity. Sheer horror."
Filmmaker Michael Moore (@MMFlint), who made a documentary about the Columbine school shooting, "Bowling for Columbine," tweeted, "Too soon to speak out about a gun-crazy nation? No, too late. At least THIRTY-ONE school shootings since Columbine."
SamEwinks (@SamEwinks), for instance, wrote, "The only gun reform we need is to allow people the ability to defend themselves, not provide nut jobs with easy targets #nra."
josefsmith2011 (@josefsmith2011) posted, "Guns don't kill people people do! Practice the 2nd amendment—keep your gun loaded with you at all times—the younger you are the better."
^^ Gun control debate erupts on Twitter | The Lookout - Yahoo! News
Saying he was “absolutely horrified” by the shooting, Rep. Jerrold Nadler said the shooting makes it clear that gun control policies in America need to be reconsidered.
“Yet another unstable person has gotten access to firearms and committed an unspeakable crime against innocent children. We cannot simply accept this as a routine product of modern American life,” Nadler said. “If now is not the time to have a serious discussion about gun control and the epidemic of gun violence plaguing our society, I don’t know when is. How many more Columbines and Newtowns must we live through?”
^^ Gun Control Debate Reignited After Sandy Hook Elementary School Shooting « CBS New York