Nothing simpler!
Taronga Zoo can be directly accessed by ferry from Circular Quay (Whart 2). If you are staying in the city centre, a short walk north will get you to the Quay (between the Harbour bridge and the Opera house). The ferry will leave at 9:15 am and it leaves every half-hour. It is a 12 minute journey and the ferry will drop you at the bottom of the Zoo.
The first ferry of the day leaves at 8:45am and the last ferry back is at 7pm. For further information on Sydney Ferries visit the Sydney Ferries website.
There are also a wide range of other public transport options to get to Taronga Zoo. It is on Bradleys Head Road in Mosman on Sydney's North Shore. Telephone +61-2-9969-2777.