Take a guided tour of the iconic Sydney Opera House or better yet, attend a performance there.
If you’re adventurous, do the Sydney Harbour Bridge Climb for the amazing views. If you don’t want to put in that kind of effort (or pay that much), you can get very impressive views from the observation deck of the Sydney Harbour Bridge Pylon.
Take a passenger ferry across Sydney Harbour and visit the Taronga Zoo Sydney to see kangaroos, wombats, koalas, Tasmanian devils, and of course the amazing platypus. If you prefer to see them naturally in the wild, you can book a tour with companies that do that. Or you can even book a hotel room at the zoo if you’re interested in seeing koalas from your balcony or you’d like an after hours tour of the zoo.
The Botanical Gardens of Sydney are very nice. Great place to do some walking and located near Sydney Harbour.
My wife and I enjoyed strolling, shopping, and eating at the Rocks Market which is open only on Saturdays and Sundays. It is located right next to the bridge and the Rydges Sydney Harbour hotel (another reason to stay there).
For astronomy geeks, the Sydney Observatory has tours and viewing through their telescopes in the evening.
Everything I mentioned above is very close to Sydney Harbour. However a popular place to visit that is outside of Sydney are the Blue Mountains.
Also be sure to enjoy formal Afternoon Tea at one of the many locations in Sydney. The Park Hyatt has a very nice one if you want views of the Opera House and harbor (with prices to match). The Tea Room at the Queen Victoria Building has marvelous ambience and will make you feel that you are in London even though it is located in the Central Business District of Sydney which means you’ll also have shopping opportunities. There are many other outstanding Afternoon Tea places in Sydney, those were the two that we have experience with. During our next visit, we may try out The Tea Cosy located at The Rocks.