With the seasons reversed, it's summer in Australia, New Zealand and the Pacific region, when it's winter in the northern hemisphere; so December-February are the hottest months (for some areas, this is the wet season too). That said, depending on exactly where you go, for parts of these areas, there is a reasonable chance of it being sunny whatever time of year you go - as a guide, Fiji has year-round temperatures of 26–31°C.
The coldest months – June, July and August - are in fact the perfect time to explore northern Australia including Cairns and Uluru, avoiding the intense heat of the Oz summer and the wet season too. Most attractions are open year-round, whether in the tropical north or the fairly temperate south. The southern coast of Australia tends to be at its busiest in January and the Easter holidays.