I loved Sydney. It just seemed to have a great atmosphere with a lot of Australian brashness about it. Very invigorating.
We stayed in a great hotel in the central Rocks area which made it very convenient to stroll down to the harbour to see the two key icons of Sydney (& Australia?) - the Opera House & the Sydney Harbour Bridge.
We took an informative, one hour tour of the Opera House, which was the next best thing to seeing a performance there.
The Royal Botanic Gardens I thought were a real gem. We even saw a Kookaburra in the wild there.
I would recommend the Hop-on Hop-off, Sydney Harbour ferry ticket. We went to Manley Beach, Watson's Bay (great fresh Fish & Chips lunch at Doyles on the sea front), & Darling Harbour. At Darling Harbour, the Sea Life Aquarium was one of the most impressive we have visited. It has some walk through tanks, where it was strange seeing sharks, stingrays and dugongs swimming round us.
From Sydney we hired a car & drove to the Blue Mountains for a couple of days to see some stunning scenery.