How long do you plan to stay in Aust?
If I had a bit of more energy, I would probably go and stay for 2 nights or so on Rottness.
Maybe a trip to start a boat trip out to the Great Barrier Reef.
Sydney is great for a city person, but if you are into nature, you would have to go out of the city, Royal National Park or up to Katomba way.
Bondi and Coogee would be worth it if you do ocean board surfing. Some even make it out to Wedding Cake Island!