Getting to Tarawa is the first real travel challenge. Bonriki International Airport (TRW) is the only international gateway and is served by a small handful of flights, all of them subject to change at short notice.
Fiji Airways flies from Nadi (Fiji) via either Funafuti (Tuvalu) or Honiara (Solomon Islands), typically a couple of times per week. Nauru Airlines runs a "island hopper" link from Brisbane via Nauru, Honiara, and Tarawa. There are also internal flights operated by Air Kiribati connecting Tarawa with the outer islands and with Kiritimati (Christmas Island) far to the east.
Schedules are sparse and cancellations or weather-related delays are common. Build a couple of extra days into either end of your itinerary and do not book onward connections too tight. Confirm reservations directly with the airline before you fly — the published schedules are not always current.