For the kids, Eddie Eagle for the young ones and good training for your 14 year old is a must. I started shooting Smallbore rifle matches when I was 13 and shooting AR15's by the time I was 15 (though I'm sure there are a lot of guys who started younger than me).
Type of Gun
As for the type of gun, give strong consideration to a handgun or a small carbine. While shotguns may get a lot of TV time, there isn't much that a shotgun can do that can't be accomplished better with a rifle, other than shooting small objects out of the sky (skeet, ducks, etc).
A simple semi-automatic handgun, with a light, will allow you to easily identify and engage a threat, while leaving you a hand free in case you need to hold a childs hand, manipulate a door, etc. Handguns are far easier to secure due to their size; currently I'm using a "Secure-It Handgun Storage Safe" to store my go-to pistol and keep my keys in it when I'm home (I live alone). If you're after extremely fast access that's child resistent, check out the Gunvault line of handgun safes. Handguns are more expesnive; I could pick up a cheap shotgun at my local gunshop for $350-$400, but a handgun (other than a Sigma) will start around $500 and go up from there.
If you insist on a longarm, look into the ar15 platform. They may look evil, but 30 rounds of 5.56mm will be just as effective (if not more so) than 5 rounds of buckshot, especially against serious home invaders. Minimum barrel length for a rifle is 16" vs 18" for a shotgun (unless registered under the NFA, approved by your local police, and with a $200 tax stamp, etc.), so it will be easier to move with inside a house. Something like http://img189.imageshack.us/img189/6053/img1921d.jpg
will cost around $800-$1000 for the base gun, $600 for the optic, and another $100-$150 for a good light, but it will be more effective than a cheap shotgun.
Also consider chemical sprays. I keep toying with the idea of replacing my carry gun with pepper spray. It's non-fatal, but getting some of the spray in someones eye will force their eye to close immediately (along with the runny nose, coughing, etc.). If you shoot someone in the heart, there will still be enough oxygen in their system for them to be functional for 10-15 second before going down; plenty of time to stab you with a knife.
Whatever you get, you need to practice with it, ideally get some professional instruction. Having a gun doesn't make someone an expert shooter, just like having a hammer doesn't make one a professional carpenter or buying a wrench giving you the ability to completely rebuild an engine. Shooting is a lot harder than it looks on TV, and having a gun you don't know how to use isn't going to do you much good.
If you decide to go with a handgun, consider getting involved in something like the IDPA. It may not be true training, but it will teach you the basics of shooting accurately and quickly. http://www.idpa.com/
Check your local laws to see if there are any legal requirements, though with kids you'll probably need to go beyond the minimum legal requirements.
As for the basement vs bedroom, in a "time is life" situation where someone just broke down your front door and is planning to kidnap one of your children, you don't want to have to run across the house, down a flight of stairs, fiddle with a combination lock, and then run back across the house while loading your shotgun one round at a time.
has some great options for handgun safes, though I've heard bad things about the fingerprint versions. I've got one of their nanovault's for my M&P Compact, which works great for me.
I'm not sure the combination locks qualify as childproof though; I know I've broken into things like that as a kid simply as a result of having a lot of free time during summer break.
Gun laws vary by state, city, and the mood of your local law enforcement. If you're going to buy a gun, get some education first on the legal requirements. In some states, you're required to run away from a home invader rather unless they are an immediate threat to your life; in other states you can shoot them at the property line. Laws for what you can own, how you have to store it, where you can carry it, what license you need, and how you can buy it vary a lot. Check out http://www.nraila.org/Legislation/State/
to get a basic run-down of your state. I'd highly recommend finding a state-level organization to join, such as Massachusetts Gun Owner's Action League, which will be able to provide you more information.
Most firearms are going to be able to shoot through walls. Keep in mind where you wife and kids are, what will happen if you miss, etc. Ideally think this over ahead of time. For an interesting read, check out http://www.theboxotruth.com/
Don't spend hundreds of dollars on a gun and neglect to have a flashlight. Most shootings ocure in low light, and you don't want to risk shooting a family member. Check out http://www.surefire.com
; their G2 LED is becoming the standard in low-cost tactical lights; bright enough to temporarily blind an attacker, tough enough to survive the recoil if mounted to a longarm, but not expensive enough to break the bank. They also offer G2Z as a stand-alone light for use with handguns, and the X300 (and X400) as dedicated weapon lights that can be mounted on either a handgun or rifle.