I used to live in Honolulu, so I can share a few slightly less-known attractions. If you enjoy hiking, consider one or more of the following: Sacred Falls, about halfway up the Windward coast; Mariner's Ridge, outside Hawaii Kai, which culminates in a stunning overlook with views of the entire eastern part of the island (you'll need to consult a guidebook to find out how to get there, though); and Pupukea Beach, way out past Haleiwa on the North Shore, which has interesting tidepools to explore. If you're slightly less energetic, consider renting a moped and riding the loop road around Mount Tantalus; there are gorgeous views all along this road (which starts at the top of Makiki St.).
Because of the heavy Japanese presence on Oahu, it's an ideal place to broaden your experience with sushi, but don't overlook some of the more typical local offerings. Ono Hawaiian Food on Kapahulu Avenue is a must. Many residents subsist on plate lunches, which usually consist of barbecued chicken or beef, macaroni salad, and vegetables such as kimchi, watercress, or corn. Or, if you're really feeling weird, go for some spam musubi (available at any convenience store) or head to Zippy's for some chili spaghetti. Really.
Budget and time permitting, if you really want to relax, go to Molokai. It's only a 20 minute flight from Oahu, but feels a world apart. There's a wonderful short hike that ends at a 1200-foot cliff overlooking Father Damien's famous (and still functioning) leper colony. Molokai also has some lovely and completely uncrowded beaches, friendly locals, and, best of all, peace and quiet.
Finally, New Year's Eve in Honolulu is unremarkable but for the overwhelmingly popular practice of setting off firecrackers. You might want to consider investing in a gas mask, unless you really enjoy inhaling cordite fumes...
Well, as they say, Mele Kalikimaka me ka Hauoli Makihiki Hou!