I'd say that Australia is a continental land mass, in the same way that mainland Europe is. However, in just the same way that the UK, Ireland, Malta, etc form part of Europe as a whole, New Zealand forms part of the continent of Australasia. Oceania is a more wide-reaching term, used to include those islands which don't really fit within Australasia.
However defining continents is never going to be easy. For example, strictly speaking, Europe isn't a continent in its own right. It's part of the 'true' continent of Eurasia.