Mickeys Very Merry Christmas Party. It's a big Christmas party that takes place at the Kingdom, it's great for kids, for anyone really.
I can tell you some about New Years. Well, I can tell you that if you want a family experience, I don't know if I would go to Epcot on New Years, I mean you can, but it's more geared towards the adult crowd (although we saw lots of kids last time we were there). They basically set up huge outdoor music stages at each country, and they play everything from country to salsa to techno. Also, because you can drink in Epcot, there is a lot more of that going on. The Magic Kingdom is probably your best bet for a "family" experience.
That being said, Epcot DOES have the best fireworks show on NYE, hands down. And they have it twice during the night, one at 10 PM and one at Midnight. Like I said, I saw lots of kids when were there 2 years ago, so it's not that you shouldn't bring kids, they'll still have fun, it's just how much of a family experience you want.