Use global roaming or buy a local SIM card. Reception varies but is available even on remote islands.
These days you'll be better off with your own mobile (cell) phone with global-roaming functionality or a local SIM. But phonecards are used in various Pacific countries – even Pitcairn Island has its own phonecards. That said, public telephones can be hard to find (and if you do find one, it often won't be working). At a pinch, you can usually find a shop owner who will let you use their phone for a local call. Some top-end hotels, however, charge steeply for the privilege of using their phones.
To call abroad from the South Pacific, dial the international access code (IAC) for the country you're calling from (usually 00 in the South Pacific, but 05 in Fiji and 19 in Wallis & Futuna), then the international telephone code (ITC) for the country you are calling, the local area code (if there is one, usually sans the leading zero) and finally the number.
For example, you are in the Cook Islands (IAC 00) and you want to make a call to the USA (ITC 1), San Francisco (area code 212), number 123 4567, then you dial 00-1-212-123 4567. To call from Fiji (IAC 05) to Australia (ITC 61), Sydney (area code 02), number 1234 5678, then dial 05-61-2-1234 5678 (dropping the zero from Sydney’s area code).
There are no local area codes in countries of the South Pacific.