Only Tarawa and Christmas Island are serviced by international flights. You can fly to Kiribati via Nauru on Air Nauru/Our Airline or to Christmas Island from Honolulu and Suva, Fiji. Fanning and Christmas Islands are popular with visiting yachties.
The main islands have airstrips and are served from Tarawa, and passenger ferries go to many of the smaller islands. You'll need your own yacht or a berth on one to reach the more out-of-the-way islands, unless you're prepared to wait a month or two for a supply ship.