Assuming you don’t have infinite time or budget (even the residents have some limits) there will be some prioritising in what you want to visit. Major landmarks are famous for a reason, and you feel foolish if you don’t see a few of them. But discovering that great little café, or obscure sight is a possibility in any destination, and its what makes a trip personal.
See the Harbour Bridge, Opera House and the Rocks area first up. They’re all within walking distance of Circular Quay. There’s also the Museum of Contemporary Art. If you get that done in less than half a day then Circular Quay has ferries and trains to take you to various other places. Luna Park and the Taronga Zoo are a boat ride across the water, and the boat ride is worth doing on its own. Alternatively, you can catch the train to Town Hall station for shopping at the Queen Victoria Building and surrounding areas.
If you want shopping and tourist sites in one location its hard to go past Darling Harbour. It has an Aquarium, the Maritime Museum, the Powerhouse Museum, and enough eateries and stores to keep you occupied for hours. It also boasts the largest Imax theatre you will ever find anywhere.
For a panorama view you can see it all from the Centrepoint tower, and enjoy the restaurant. For the more daring there is the Harbour Bridge climb, an exhilarating but demanding endeavour. Those on a budget might prefer to walk the bridge at street level.