I think there should be a free right until you commit a felony, DUI= no gun, beat up your wife/children = no gun, Even been on schizo therapy= no gun, Smoke crack meth etc = no gun.
But other than that a simple id check should be the only requirements for a average gun (average gun= anything the cops use, including swat, no bazookas or anything like that). Gun stores should always have a list of the people that are prohibited to have guns so that the employee can make a deeper check if the pictures match.
I think there should be a(elective) course in high school where you can learn firearm safety and doctrine and during those classes you can also weed out the schizos.
Also military service at the age of 18 too weed them out even more.