Just about every local bar and nightclub in LA will be hosting a New Year's Eve Party, but for the grandest of them all, check out my guide to the Biggest New Year's Eve Celebrations Around LA. There are events listed for every budget. Ticket prices on some events go up closer to the date and some events sell out, so get your tickets now.
The AAA usually offers Tipsy Tow service on New Year's Eve. If you're too intoxicated to drive home, you can call for a Tipsy Tow and the driver and car will be driven home up to 7 miles for free. Beyond 7 miles the driver will be charged for the tow. Passengers are on their own. Call 1-800-400-4AAA.
If you don't have a designated driver, consider taking the Metro. The press release hasn't come out yet for this year, but usually Rides are free on New Year's Eve from 9 pm to 2 am.
You might also consider catching a ride from Uber. I'm not convinced that they're more economical than a standard taxi, but it's easy to book and pay with a smart phone. Beware, however, that Uber's rates skyrocket during busy times, so make sure you read the rate you're agreeing to before you book. LA City Cab also has a iPhone app, but I think it's only for booking, not paying.