Take a lunch cruise on the harbour. It's fabulous. Also if you're not scared of heights, you can do a bridge climb over the top of the harbour bridge or venture out onto the roof of Centerpoint Tower. If you don't fancy the rooftop, the observation deck has 360 deg views of the city & they also have a revolving restaurant in there. If you go there, make sure you book a window table for maximum views. Same with the harbour cruise.
As MumtoEllie said, Darling Harbour is a good spot. You'll also find the aquarium there. Circular Quay is where you'll find the opera house & is also where all the cruises & ferries depart from. Personally, I wasn't that impressed with Chinatown but everyone's got different tastes.
I'm hoping to get down there later this year to catch up with some friends. Top of my list is the zoo then the bridge climb & the roof of the tower.
Sarah's the resident Sydney expert around here. She can tell you a lot more.