Really depends on what your interests are as to where you go and for how long, but as a general guide if you want to miss the more warmer humid weather, keep heading south though there are some island trips you can do from Brisbane - Moreton Island for dolphins, snorkelling and sand duning, and North and South Stradbroke Islands too, all having great beaches and a relaxed environment.
Sydney will likely be on your agenda, and heaps to do and have a look at there and also some reasonable places to stop at like Byron Bay, Coffs Harbour, Port Stephens with many smaller places in between.
If you have not booked accomodation for Sydney between Xmas and NY, you ought to visit outside of those times for it'll be near impossible to get decent accomodation, and also difficult if not likewise for most coastal places.
Have a look at the Destinations and Hostel guides for more details and transport re how to get about and again depending on just how much of Australia you want to see will determine how much of NZ you could also see but three weeks for both is going to be a bit of a crush.
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