Hope this is okay to post but here are just a couple of suggestions of mine -> http://bittenbythetravelbug.com/the-day-richard-branson-looked-at-me-and-i-ran-away-twice/
I'd also recommend going to check out Glebe, which is quite a trendy suburb in the inner city and visit Glebe Markets. :) about 6 years ago
Take a walk along the pathway from Bondi Beach to Bronte via Tamarama (city buses are at all beaches) it's awsome.
An early ferry to Balmoral for breakfast, take a zoo link bus back, spend a few hours then a ferry back into the quay. From the ferr terminal venture through the Rocks area.
Kings Cross has markets in the plaza on the weekends, good people watching!
The Botanic Gardens, wonderful views of the key harbour sites.
St Marys Cathederal, just divine. about 6 years ago