If you are bored out of your brain (and have lots of money – or conversely, very little money) then Darling Harbour is for you. There are LOADS of things to do. Shopping, sightseeing, the aquarium, iMax, the harbourside, the exhibition centre (normally has something on), Star City (just behind the harbour), the Chinese Garden of Friendship, Powerhouse Museum, etc. I dare you to be bored there!
When it opened up (after a multi-million dollar redevelopment) in 1988 (as a present to the Sydney population for the Bicentennial no doubt) I had been a regular visitor right up until I moved to the UK in 1999. When I DO go back to Sydney, it is one place I always visit along with Sydney Harbour itself.
A ride on the Monorail is a must too!