Nunchuck is the add-on to a Wii Remote to give your left hand additional buttons and an anolog stick. I would recommend getting one for each Wii Remote that you get. Most games (at least Wii games) will want you to have one.
The Wii U GamePad will last about 3 hours or so before it needs to be charged. It can be played while plugged in to give more time, but you lose mobility, so some games cannot be played as designed when tethered to a power source.
Wii Remotes (not Wands, that's the Playstation Move's lame term for their glowing lollypops) use 2 AAs batteries by default and will probably give you 10-15 hours of use per set of batteries. I've used a recharge station in the past. While it was nice at first, over time, the batteries lost maximum charge and when it got to the point of only lasting about 30 min, I threw them out and went back to using AAs.
You do not get any Wii Remotes with the Wii U, and each additional will cost about $40. I would recommend you find a copy of Wii Party. It is $50 and comes with a remote, so you are effectively getting a game for $10, which is very good. Your kids will enjoy it. But you will need to have a Wii Remote for all of them if they are to play multi-player (as the game is designed to be played).
If you are a fan of traditional controllers, you might want to consider the Wii Pro Controller (or the Nyko equivalent if you think the Pro Controller is too large and/or don't like the button placement -- like if you are used to a 360 controller, etc.) I don't think the kids would get much use out of that, it would be for your use, so you wouldn't want to get more than one of them initially.