Lots of good advice for you so far Marine. I am a father of 2 small boys and as much as I love both of them I know that they are not ready for unrestricted access to firearms.
I have a presentation that I give on firearm safety in the home. I use it for training other physicians on how to counsel their pro gun patients. (Patients who would react negatively to the party line of "No guns in the home")
As you mentioned there is a continuum between safety from outside threats and safety from the dangers of improper firearms handling. It starts with a threat assessment. Clearly there is a non-zero possibility of your home being targeted for home invasion. The first step is to minimize the likelihood that you will be targeted. Cutback your shrubs, plant and cultivate unfriendly shrubbery, add motion lighting, buy a dog, an alarm system or both. Do not advertise your wealth. Dispose of boxes from high value purchases discretely. Park your cars in the GARAGE.
Whether we like to admit it or not, once firearms are inside the home, in most situations, unrestricted access by untrained persons to firearms and their mishandling becomes a greater threat than criminal attack. I understand that this is a debatable point. (please note that this is a VERY qualified statement)
The dangers of firearms can be mitigated through safe storage practices and training in the proper handling of firearms. For adults training in a basic firearms course is a minimum. A course like the NRA's Personal Protection INSIDE the home is better. For Children Eddie Eagle is a very good option. Once children are responsible and reliable enough for range training (Your call Dad!) they should be treated like adults and trained accordingly.
Even (sometimes ESPECIALLY) trained children should not have unrestricted access to firearms. As from a developmental standpoint their ability to exercise good judgment and understand consequences has not yet fully developed.
Most firearms should be stored unloaded in a proper gun safe. Although a simple locker can be used it is simply not as secure as a modern, fire retardant gun safe. Most families should already have some form of fire resistant storage to protect their vital documents anyway so a good gun safe does double duty and is a good value. Ammunition should be locked away separately in another container. Lockers or safes for ammo each have their advantages. A proper ammo safe has venting as well as fire resistance to prevent the safe from becoming a bomb in a house fire and is the most secure option. Lockers do not require venting and are significantly cheaper. Most ammo will "cook off" a round or two at a time and it is only the brass that needs to be contained as the bullets tend to stay put. (it's a momentum thing) This fact can make a locker style storage container the best option for storing small amounts of ammunition.
Lastly we come to the concept of "ready guns" here I am going to advocate a position which has ruffled some feathers in the physician community: Concealed or open carry in the home for BOTH adult partners. Think about it, if your pistol is on your person, you know beyond a shadow of a doubt that they are not accessing it. Clearly this needs to be done carefully, I do not recommend this for everyone. If you and your partner are prone to argument or your marriage is in trouble this is not a good solution. Similarly if your teenager is driving you out of your mind, it might be better to take the gun out of the situation. This takes some responsibility on your part to judge if this is wise for your own domestic situation. No one else can tell you if it is a good idea.
A rapid access safe with either coded or biometric access is essential. This is where you store your firearm when you are not carrying it. It is stored loaded and ready. This minimizes "handling" of the gun when holstering it and allows it to be ready should emergency access be required. There are even rapid access options for the storage of long guns.
This has been a long post I have put forward some controversial information here. Pro gunners will not like my assessment about the dangers of unrestricted, untrained access to firearms often being a greater threat than home invasion. Anti gunners will balk at the carry in the home bit. I have always felt that a good compromise makes everybody equally angry. I hope I have achieved my goal.
"Get this through your head! We're not fighting to have everybody think the way we do, we're fighting so that people can think whatever they want! Even if they don't agree with us!"--Stalker, GI JOE #39