My knee jerk response was a 12ga pump but after thinking about there being kids in the home I was leaning more toward a rifle or a handgun.
With kids in the house, why are you more concerned about the shot gun than the rifle?
It's a matter of preference.
You can't answer the door with a shotgun, but a handgun can be out of sight and quickly brought to bear.
Sometimes we address the subject as though we can only have one or the other. I have both, as do many.
A pistol may be all that's handy in some emergencies (but only if you see that it is). When we retreat to our bedroom, it may be shotgun or .223 time.