crosvs,
Agree with Xfoomph that Taronga is worth a visit. Featherdale ia very popular with tourists and many buses come and go every day. Last time I went (some years back now) there was definitely opportunities to get in a meet and greet animals and touch a koala if you wish. Not sure where you are staying but it is about 40 minutes from the centre of Sydney.
Also agree with onions that the bridgeclimb is great. Not cheap but something you will remember. Wouldn't waste time going to Melbourne though
... Just kidding, friendly interstate rivalry. Melbourne has some outstanding restaurants and cultural aspects!!