Australian seasons are the opposite of those in the Northern Hemisphere—when it's summer in the north, it's winter south of the equator (Australian winters are fairly mild but a bit rainy). Our favorite months are October and April, though September-May is all right in most parts of the country. December and January are the hottest months, July and August the coldest.
The Great Barrier Reef is best visited June-October: there's low rainfall, the temperature is comfortable during the day, and the water visibility on the reefs is at its best. Cyclones can disrupt sightseeing plans November-April and dangerous marine stingers congregate in the waters of the far north November-May.
Take a sweater, no matter when or where you're going.