If your ears have perked up, note that while Kiribati is definitely worth a once-in-a-lifetime visit, it's not going to be really easy to get there. Your best bet is to head down to neighboring Fiji and hop on a Fiji Airways flight from Nadi to Bonriki International Airport, which is considered the gateway to the rest of the nation's islands.
Pack light (literally, as it's going to be in the 90s), and prepare yourself for the sticker shock of the airfare. It's $690 USD roundtrip from Nadi to Kiribati, leaving on December 29, and that doesn't include the flight from LA.
Even if you can't head to this remote destination for the new year, don't knock it down too far on your bucket list; Kiribati is one of ten nations most susceptible to the effects of global warming. The island chain sits at an average of 2 meters below sea-level and is quite vulnerable to rising sea waters.