Go on the ferry to Manly - only takes about 30 minutes to get there and theres lots of nice places for lunch, beach etc - wicked place loved it!
Are you going to have a car or will you be public transport? If so there is a special card that you get at the airport and its an all day pass for 7 days (we got it even though we were only there for 3 days) and it took us everywhere we wanted to go, even includes the ferry ride to Manly, Luna Park etc, to Bondi - everywhere it was awesome.
The aquarium was awesome, quite expensive - but then again most touristy things are arent they!
Darling Harbour has nice gardens and a playground there.
I just love love love love love Sydney!