I kind of agree with Sbryant. I too was introduced to firearms at a very young age, probably around 5 years old. I started shooting an old bolt action .22 rifle that my dad's dad gave him when he was young. Moved onto a Ruger 10/22 and .22 pistols. My uncle has been into collecting firearms for as long as I've been alive, and he has always had ALOT of firarms (I can probably thank him for my sickness/firearms collecting). My brother (younger), Dad, and uncle spent many hours together, needless to say, I was exposed to a wide variety of firearms as a kid. My brother and I were around WAY more firearms than any of our friends. We too were taught the absolutely necessary rules of firearms. like Sbryant said, never point a firearm at something you are not willing to KILL, and always keep the weapon in a safe condition until you are ready to aim and KILL what you aim at, always treat all firearms as if they are loaded (I could go on and on). Whatever you do, MAKE SURE you have a secure place to store your handgun when it is not on your hip.
Anyway, I don't think there is any reason for concealed carry to be a big secret. I carry a handgun everywhere I go, my wife knows, and she knows what to do when I tell her to grab the kids and get away from me. My kids know I carry a handgun, although they are still a bit young to really understand it. For me it is easy, but it is part of my job to carry a handgun, and I tend to see things pretty black and white.
Ultimately, this is a decision you have to make, I don't know your situation, the community you live in (political climate), or what your family thinks about firearms. I just don't think carrying a handgun is something you should keep secret from your family, it may be very important for them to know.