Buy a zoopass from Circular Quay and go to the zoo via ferry. The pass includes your round trip on the ferry, a cable-car ride to the top of the hill (or you can take the bus), and admission to the zoo.
While I was there they were working on a new enclosure for the elephants, and it looks like it's going to be wonderful. Should be done soon, too.
Great views of the city from some of the exhibits, particularly the giraffes and mountain goats. The free flight bird show featured mostly native birds.
Lots of Australian animals on exhibit. This is the place to go if you want your picture taken WITH a koala. You won't actually be holding one, but it will be in the frame with you ... and it's the best price you are going to find for that sort of thing.
Allow 3-4 hours for this, and take your time.