Kiribati has two international airports: Bonriki International Airport (TRW) on South Tarawa, and Cassidy International Airport (CXI) on Kiritimati (Christmas Island). They are not connected to each other by international flights — you have to fly in and out of each separately.
Fiji Airways is the main carrier. It flies Nadi–Tarawa twice a week (Mondays and Thursdays) and operates a Honolulu–Kiritimati route weekly, departing Honolulu on Tuesdays and returning Wednesdays. This makes Honolulu the most convenient gateway for Kiritimati.
Nauru Airlines operates two weekly services into Tarawa from Brisbane, routing via Nauru. One service continues onwards to Kiritimati, Majuro, and Pohnpei, making it useful for island-hopping across Micronesia and the central Pacific.
Air Kiribati is the domestic flag carrier and handles inter-island flights within the Gilbert Islands. It does not operate international routes.
Non-US citizens flying to Kiritimati via Honolulu may need a valid US visa for the transit. Check requirements well in advance — the US transit visa rules apply even for brief airport connections.
The departure tax is included in most airline ticket prices, but confirm at time of booking.
There is no regular international passenger shipping service to Kiribati. Occasional cargo vessels call at Tarawa, Banaba, and Kiritimati, but these are not a reliable way in for most travellers.