I'm Australian (Melbourne) and have spent about 18 months of the past 2 years wandering this country!
Comments:

• Weather is better later in year. Winter here is nothing like your winter.
• Can still swim in northern Queensland waters in September. Sadly the coral is bleaching on GBR with effects of climate change. Ningaloo Reef in Western Australia is better now (next trip!). Can see lots by snorkeling.
• Restaurants, theatre etc is best in Melbourne. Terrific Botanic Gardens also
• in Victoria don't miss driving along the Great Ocean Rd (there are many companies who do day trips from Melbourne although I'd take at least 2 days). Wilson's Promontory is also a favourite of mine.
• Kangaroo Island off the South Australian coast is fabulous for wildlife, rugged beaches, and food. Can fly there.
• Tasmania - MONA in Hobart, food, beaches, Maria Island, Bruny Island. Tends to be cooler and probably not swimming weather during your dates.

3 weeks is being ambitious. Bear in mind the distances. Remember how much time is lost as soon as you involve an airport (even in Australia). The Great Barrier Reef is best accessed from somewhere like Cairns (which is 3.5 hrs flying time from Melbourne) or Port Douglas.
Whatever you decide, you'll have fun!
Consider Melbourne with day trips, Hobart with day trips, Kangaroo Island.