Centennial Park in the middle of Sydney is easy to do for birdwatching. The central lake has islands which attract egrets,herons etc.
The Royal National park is pleasant to walk around. It's not far from the centre of Sydney.
You can get the ferry to Manly. Turn right thru the shopping centre and there is plenty of good scrubland to explore.
You'll see a lot of birds just walking thru the suburbs. We stayed in Waverley where you can access the coastal path-saw some decent birds there.
A trip to the Blue mountains is usually on the agenda so it might be worth trying.