See the country articles for detailed information on how to Get in.
The major countries of Australia and New Zealand do of course offer connections from all continents, although there are few direct flights from South America. There are some other gateways offering other opportunities to get in to Oceania, and for interesting itineraries. Air France connects New Caledonia direct with Tokyo and Paris and also flies to Tahiti. Onward connections to Sydney and Auckland are possible. Air Pacific connects Fiji with Los Angeles with connections through to Sydney, Auckland and Tahiti. Tahiti is connected to Los Angeles, and you can fly to the Cook Islands direct from there. Air New Zealand provides a service to Tonga and Samoa from Los Angeles and Auckland. The Los Angeles service is subsidized by the New Zealand government as a form of aid to the the two countries. Manila, Guam and Honolulu offer a gateway to many countries of Micronesia, mainly on Continental Airlines.
The smallest islands with less tourism present additional challenges to get to. Many are entirely deserted, and some have restictions on access.