Assuming that you wan't be venturing too far from Sydney, I would recommend the following:
1) Climb the Harbour Bridge (tours done by BridgeClimb).
2) Visit Taronga Zoo. It is world class.
3) Sky Walk from Sydney Tower (above Centrepoint shopping complex).
4) Ride a jet boat around the harbour - not for the faint hearthearted.
5) Bushwalk in either the Mt Kuringai National Park (north) or the Royal National Park (south).
6) Eat Asian cuisine at the Malaya, downstairs on Castlereagh Street. Best Laksa in Sydney.
7) See a movie at the IMAX giant cinema.
8) Scuba dive near the entrance to the Harbour (Sydney Heads).
9) Surf at Manly, Bondi or Narabeen beach.
If you're willing to drive some distance:
10) Visit the Blue Mountains (west), the Three Sisters in Katoomba, go on the skyrail.
11) Plenty of paintball companies running games in the bush around Sydney.
12) Get attacked by snarling, vicious pelicans at The Entrance (north).
13) Take a motorbike ride up and down the Southern Highlands. Some great windy roads down there.
There are probably more but it's my bedtime and I need my beauty sleep.
Cheers.