Why have a party of course! NOT! Dogs don't tend to like loud music, fireworks and rich food and they certainly will not be at risk of drinking and driving. Want to spend a fun New Year's with your dog? Try using your day off to take a long walk in the woods (also good for working off a hangover by the way) If you live in a temperate place or were lucky enough to take a vacation someplace south, take a walk on a beach, or just go out and toss a toy. For Northerners, if you have snow on the ground try snow shoeing, cross country skiing or get a light sled and let your big dog take you for a ride. You can also curl up with a good read by a toasty fire and have your dog warm your lap or your feet. They like quite time too…especially if you are with them. Dogs are very adaptable and can be happy doing anything or even nothing as long as they have company.
If you are looking for a good read about dogs, I just finished The Genius of Dogs by Brian Hare and Vanessa Woods http://www.amazon.com/The-Genius-Dogs-Smarter-Think/dp/0525953191. This book does a great job of blending science and dog lore into a comprehensive understanding of how dogs are smart and how their intelligence compares to other higher mammals.
You can also try your hand at making home made dog snacks. The ingredients are typically easy to find and if you worry about your waistline when you bake, then focus on doggy snacks instead.
Regardless of what you do on New Year's day, take a moment to be thankful for friends and family and especially for having the joy of a dog in your life.