Note: I, myself, do not have children. My wife reminds me of this fact every time there's a baby in a store. (As if I'd forgotten: "Oh, man, I always thought we had one, but turns out ... that's a chair.")
One of the folks I game with here has kids (2, I think, or 3) and I know at least the oldest two game. They're all very normal people, not "geeky". I'm not sure how the kids came into gaming, but I know their dad has been gaming for quite a while, since AD&D 1e at least.
Of all the other gamers with kids, they're the only ones that I know of who game regularly. Most of the others, though, are teen agers ... and anything you want your teens to do, they're not going to do, because they're teenagers.
But your kids' interests will be naturally shaped by YOUR interests, and how you TREAT your kids. They're impressionable, but they're little people, as well. If you try to "get them into" gaming, you might make it something they think they need to be good at, will get intimidated by, and probably not like.
Think about how you, yourself, got into gaming ... what did your interest grow out of? Reading, fantasy, etc. If you read to your kids, they'll probably read, if they have access to the gaming, they'll probably get curious about it, especially during the times in their lives that they want to be just like their parents. Because you game, they'll probably game.
Plushy dice.
--fje