If you go to a park on New Year's Eve, PLEASE do not expect to actually do much. Go to enjoy the atmosphere and the celebration of a new year and maybe get in a ride or two. If you go expecting to visit the parks as actual theme parks you will be sorely disappointed and exhausted. I also recommend packing snacks and a lunch.
Lines will be everywhere in the parks and they will be long even to use the restroom. Some parks like Magic Kingdom will most likely close due to capacity. Other parks like Epcot will do their best to keep shoving them in. Hollywood Studios will reach capacity for a couple hours, but will reopen back up, and Animal Kingdom is a toss up but usually the least crowded.
If a ride typically has a 30 minute average wait through the year on a moderate crowd level day, expect the line to be at least 90 minutes if not longer. If you are used to waiting 5 minutes in line to order lunch, expect to wait 30 minutes. If you expect to grab a spot for a parade 45 minutes before showtime, expect to stake a spot an hour and half before hand.
New Year's Eve can be one of the most exciting times at a Disney theme park, but if you are not prepared for MASSIVE amounts of people, crowd control issues, extremely long waits even for unpopular attractions, and no plan to tackle it all, then you will not enjoy it and will never return for New Years Eve.
Oh and best advice I can give you when traveling through crowds as a family. Before venturing through a crowd ALWAYS discuss a meeting point if you become separated. Don't expect to be able to stop and wait for them to catch up and don't rely on your cell phone. If you are at the castle and are exiting the park, agree to meet up at Exposition Hall if you get separated on the way. Just focus on getting there and not getting there together.
Hopefully, it won't detour you from having an awesome trip, but I figured someone needed to say it like it is.