1. Go to the top of Center Point Tower (City)
2. Visit The Rocks - next to Circular Quay
3. Take the Manly Ferry - Circular Quay to Manly and back
4. Drive out to Bondi Beach - maybe have a meal at Icebergs (nice view over the beach) or Sobo. Drive up to South Head from there.
5. Have Fish & Chips at Doyles on the Wharf (Watson's Bay) one evening - can watch the sunset over Sydney
6. Drive out to North Head (near Manly) one morning - great views back into the harbour
7. If you have a car, you can drive this route (anticlockwise to avoid tolls) and go over 5 bridges:
http://maps.google.com.au/maps?f=d&s...=UTF8&t=h&z=13
8. A trip to Taronga Park Zoo is worth while - you can actually catch a ferry there from Circular Quay
9. There are also Harbour cruises that are quite good - again, most depart from Circular Quay
10. If you are keen on driving, a trip out to Katoomba (Blue Mountains) could occupy a day
11. For other dining-with-a-view I can recommend Aqua Dining (Milson's Point - right under the northern end of the bridge) and Altitude (in the Shangri-La Hotel in The Rocks).