Australia and New Zealand are the best places to work and or volunteer in Australasia. Australia in particular pays very well, even for minimum wage jobs, so many backpackers settle down for a couple of months to reinvigorate the budget. The most commons types of backpacker jobs are menial, like bar work or farming, though if you have a professional qualification you can get white collar jobs which pay much better.
It will be extremely difficult to find work anywhere else in Australasia due to the small populations – there isn’t much call for backpackers when there are more than enough locals to fill positions. If you do find employment on one of the islands it will most likely be unofficial – i.e. cash in hand – but don’t bank on it - literally.
Due to its main areas being quite affluent, Australasia isn’t a classic volunteering destination, though there are opportunities to do so in Australia and New Zealand, particularly with animals and the environment.