Wow! I was in Sydney last week - OK, didn't actually get off the plane, just sat on the tarmac after offloading a sick passenger....
Anyway, I grew up in Sydney and this is what I'd recommend:
Toronga Zoo and you can get in an enclosure with the Koalas and Kangaroos
Climb the Sydney Harbour Bridge
Brunch down at Bondi Beach, followed by a swim!
Pitt Street Mall....and DFO (Direct Factory Outlets)
Tetsuya's if you're after a fabulous, special meal...yum
Snooze on the grass at Hyde Park
Lunch down at the Sydney Fish Markets
Bondi Junction and Chatswood has FABULOUS shopping
Not sure what the situation is there after all the bush-fires, but a day-trip to the Blue Mountains is really pretty.
Have a great trip! Australia is such a beautiful country!