Roughly 99.9% (+~0.1%/-~.2%) of so called assault rifles in the US are used for target shooting or varmint/small game hunting. Contrary to an often expressed opinion, these are legitimate civilian uses. By current reports, the Santa Fe High School killings used a revolver and a shotgun (possibly shortened illegally), rather than an assault rifle, however defined.
Roughly speaking, the US media and those who refuse to accept the plain historical meaning of the second amendment (and the corresponding part of the English 1689 Bill of Rights) classify as "assault rifle" any self-reloading rifle that uses detachable magazines and has a pistol grip or any of several other mostly cosmetic characteristics that make it look like a military issue weapon. The most common variants, chambered for .223 or 5.6 mm ammunition, generally are unsuitable for hunting large game like deer due their low power, and are prohibited for that use in a number of states. At close range, like many other firearms, they make terrible wounds sometimes described in gory detail in articles, but they probably are not materially different from other weapons that deliver comparable kinetic energy. Limiting legal magazine capacity might have a minor effect, but it is worth keeping mindful that with practice it takes under two seconds to change a magazine. Banning these weapons is largely a waste of effort.
Meaningfully licensing gun ownership, with a similarly meaningful requirement for safety training, is unquestionably a good idea, but is unlikely to affect the occurrence rate of these awful rampage shootings other than, perhaps, making those who undertake them more skilled at their self-assigned task. The result of safety training failures is, if I remember correctly, in the order of 500 per year, some of which would occur even with a strict, frequent, and well-implemented training requirement. Classroom instruction, no matter how thorough, tends to evaporate over time.
Gun safes are widely used by responsible firearm owners, apparently including the parents of the of the Santa Fe High School shooter, and are, I think required in some jurisdictions. Biometric locks, suggested in one post, might be helpful, although there are numerous reports of methods by such things they can be and have been circumvented. It is not clear that more effort in this direction is likely to be very beneficial.
More extensive and thoroughly implemented checks preliminary to buying guns also is a worthy effort, but probably wouldn't improve things by much. Like credit checks, they will yield nothing for individuals with no history, and will not prevent those with no known history of bad behavior from acquiring weapons under whatever other control regime may exist. In the context of the US constitution, there also are due process issues related to abridgment of rights. A ban on firearms acquisition by "mental cases" or those on terrorist watch lists would be likely to be overturned by the courts unless backed by a requirement for specific evidence for each case. Psychiatrists also would object on the grounds that the great majority of their patients are not a threat to themselves or others. And while they generally are limited by professional confidentiality standards, they may and in some cases are required to report dangerous patients, but that is a judgment based on inexact knowledge and certainly will miss some cases in both directions.
To a first approximation, there are somewhat more than 30,000 firearm deaths annually in the US. About 2/3 of these are suicides, which some would argue exercise a fundamental and inalienable human right. Around 500 are accidental or by children playing with unattended or "found" weapons. Of the remaining 9,500 - 10,000, the great majority are related one way or another to what might be considered "ordinary" criminal activity of various kinds, including some family blowups that sometimes are classified as "mass shootings".
A small number, probably averaging around 100 - 200 per year, can by some standards be reasonably placed in the category of these school killings, the Las Vegas massacre last year, and others. It is not clear that these can be ended or significantly reduced by any actions short of repealing the second amendment and enacting seriously restrictive laws that result in confiscation (presumably with compensation) of nearly all privately owned firearms. In the US that is unlikely to happen any time soon, if ever.