Before you start immortalizing your holiday gatherings of friends and family, learn to use your camera. Celebrate the new year by taking it off auto. Party in a priority mode.
That's the most important piece of advice I can think of for any photographer. Whether it's a cell phone, a point-and-shoot, or a power dSLR, if you want to take better photos, you have to start by taking at least 30 minutes to understand your camera's controls. Otherwise, you're going to have nothing but bad shots at best and missed opportunities at worst. I find shooting with a strange camera is like driving a strange car in the rain: don't wait till the downpour before looking for the wiper controls. Editors' note: This was updated December 16, 2011, to include more tips; now there are 17!
And I say that from experience: I went into this past Thanksgiving with a completely unknown camera, the Panasonic Lumix DMC-GX1. Which is why I start my tips off with shooting raw+JPEG if your camera supports it. Raw is your safety net for all the underexposed, incorrectly white-balanced, high-ISO-sensitivity shots that you take by accident when shooting with strange equipment.
There's always something you can do to improve. If you usually shoot on full auto, switch to a program or semimanual mode and see how having more control allows you to produce better images. If you have full command of your camera, think about your photographic style and how to change or enhance it. And if you've got style nailed, try something new. Configure a camera for remote shooting or time-lapse in the kitchen or living room. Set up a tripod in an out-of-the-way room and practice portraiture on relatives you see once a year. Start thinking now. It'll pay off later.