Are you planning on going to Wildlife Sydney Zoo next to the aquarium at Darling Harbour, or to Taronga Zoo? I haven't been to Wildlife (I think we are going in April), but enjoy Taronga. There is also another aquarium over at Manly, but we prefer the Darling Harbour one.
We enjoy walking around the city. We've had fun in the past doing the Tank Stream multi geocache (GC1CR58), if you 'cache. :) Downstairs at the Post Office was my favourite segment of that cache.
For a nice, long walk around you could start in Hyde Park, enjoy the avenue of trees and check out the Australian Museum nearby ($15/adult). St Mary's Cathedral and Hyde Park Barracks are nearby. I like to head down Macquarie St and check out what is happening for free at the State Library of NSW. Either way, you could then do a nice walk through the Botanic Gardens, check out Mrs Macquarie's Chair, walk along the water around Farm Cove (read up on some of the early settlement history, and imagine it 230 years ago). Check out Government House (in the gardens), then head on to the Opera House. Walk from the Opera House, past Circular Quay (looking out for the markings on the ground that note the 1788 shoreline). Walk from Circular Quay onto the Sydney Harbour Bridge (looking at a map, we seem to have accessed it via the stairs on Argyle Street, near the corner of Cumberland St.) It costs money to to the Bridge Climb across the arch of the bridge, but it is free to walk across the regular part of the bridge.
After crossing the bridge we like to take the underpass at Fitzroy St and make our way down Alfred St South, back towards the harbour. You get some beautiful views, and go past the North Sydney Pool, which is a bit famous as you can do laps with views of the bridge. You might enjoy continuing around to Luna Park and having a walk through. It is free to enter, but you pay for the rides, etc. Maybe get someone to take your pic inside the big mouth. :)
Then the Milson's Point wharf is nearby. We like to then take the ferry to Darling Harbour. This will bring you close to the aquarium and Wildlife Zoo. We like to walk past the Chinese Garden of Friendship (must go in one day), then make our way under the overpass, past the old entertainment centre site and towards Paddy's Markets at Haymarket. From Paddy's Markets we like to enter Dixon St (the view is better from this end of Dixon) and get Chinese food. :) Around here is known as Sydney's Chinatown.
From here you are only a few blocks from Central Station, to return to where ever you are staying. :)