Because there is so much to see I'll spread this post out over a couple.
Lets start with getting there...
Sure you can drive to the Zoo, head north over the Sydney Harbour Bridge and battle the traffic down Military Road through Neutral Bay to Mossman, or, you can catch the ferry!
Catching the ferry from Circular Quay has to be everyone's favourite way to get to the Zoo! Its fairly inexpensive and you can bundle your zoo ticket with your ferry fare. There is no better way to see the world famous Sydney Harbour and its icon land marks, The Sydney Opera House, The Sydney Harbour Bridge and Luna Park. The ferry trip takes about 20min (don't quote me on that its a rough estimate) and takes you past Mrs Macquaries Chair, Woolloomooloo Navel base and Fort Denison. Have your camera ready as you will be clicking the whole way!
When you alight the ferry at the port at Taronga Zoo, most people (if its included in their pass) will catch the cable car to the top entrance. Although there is a massive hill you can walk up if you like the exercise!
