I live in Australia, and though I have not been to Sydney yet (well, one time I sat in the airport) I can vouch for a few places being worth your time:
Head up to Avalon in Sydney's Northern Beaches. Otherwise throughout the city Surry Hills is a great place to grab coffee, Newtown for interesting people and lifestyle, Circular Quay is beautiful anytime of the day and Glebe markets on a Saturday or Bondi markets on a Sunday are always worth a look. Shopping-wise, The Strand arcade on Pitt Street is a good start if you're looking for Australian designer shops, and Oxford Street has a lot to offer as well.