I've been to WDW plenty of times between Christmas and New Year's. You can't overestimate the size of the crowds. During the middle of the day at the parks, the crowds are ridiculous. There's little reason to be at the parks between 11 AM and 4-5 PM or so. The lines for everything are enormous. It's much too aggrivating to be enjoyable.
I've found the best strategy on days you're going into the park is to divide your day into quarters. Sleep for the first quarter, wake up early and get to the park 40 minutes before official opening. Disney does often open the park earlier than they say they will. Leave the park after a few hours, no later than 11, and sleep. Go back in the evenings and go on what you missed during the morning. If you do it this way, you should be able to see what you want with as little hassle as possible.
As others have said, if you want to eat anywhere that requires a priority seating, make the priority seating as soon as possible. Don't expect to walk up to the park and get a meal at any of the sit down restaurants that evening. Even if you're not interested in eating at a table service restaurant, I'd recommend it over counter service. The lines for food are longer than the lines for many attractions and it will take you just as long to find an empty table.
On New Year's Eve, don't expect to get on mnay attractions. Everything has ridiculously long lines, except for El Rio Del Tiempo. (quick digression: a couple of years ago it was pouring on New Year's Eve. There was still a big crowd at Epcot waiting for the first Illuminations show, however. We headed into Mexico to avoid the rain and kill some time. Everyone else had the same idea. The entry portion with the art exhibits was filled wall to wall with people. It took 15 minutes just to get into the shops area, which wasn't any less crowded. El Rio Del Tiempo? No line. We got our own boat.) I've had the best time blowing off the morning and showing up at Epcot late in the afternoon and just wandering around World Showcase. It's also a good place to go if you have teenagers, since they have about five DJ booths set up around the park.
New Year's Day is a great day to go to the parks if you don't want to stay out late on New Year's Eve. In my experience, Epcot has been fairly empty New Year's morning. You should be able to easily make your way around the big attractions in Future World before World Showcase opens and then have World Showcase to pretty much to yourself until the afternoon.
