Other places I highly recommend in Sydney:
These are all beautiful places to visit, especially if one appreciates nature. There is also one place I can't believe DMan missed: that's the Royal Botanic Gardens area. There you can find the domain, the Art Gallery of New South Wales and the beautiful Mrs Macquarie's Chair, where you can dangle your feet in the harbour, with a stunning view of the Sydney Opera House, the Sydney Harbour Bridge, North Sydney and the CBD.
For people who are young and hip, Newtown and Glebe are must-go places. This is where all the bright young things hang out, but they're also cool places to visit. Newtown is full mostly of University students and can be compared with, say, the Village in New York.