Aside from shopping there are so many cool things to do with kids, Sydney is really kid friendly. The Toronga Zoo is a must and it's awesome just getting there via ferry and gondola. The aquarium looks great but unfortunately we ran out of time. But the Darling Harbor playground is amazing, so many things to do there but make sure you take a change of clothes!
One thing I love about Sydney is that all the cabs have car seats so you don't need your own car if you are staying central. Every cafe and restaurant we went to was totally cool with kids. Our favorite cafe had to be Reuben Hills, I can't stop thinking about the Strawberry, Rose & Vanilla smoothie!!!