I "studied" abroad in Sydney in college, and then DH and I lived there for a few months when we were first married.
Even in the winter, Sydney is pretty mild. And around Bondi there are great walking paths that meander along the coast. As niccig said, there are lots of great cafes and restaurants with a patios - the Thai food in particular is amazing - every little hole-in-the-wall is great.
Public transportation is pretty good - buses and trains and ferries. Think about going to the harbor and taking a water taxi across to Doyle's for seafood - really fun.
The acquarium is of course incredible - will be great with a toddler.
The Paddington Market on Saturdays is also great for picking up unique souvenirs, and easy to wander around with a stroller.
Also as niccig suggested, the Blue Mountains would be a great day trip. The Hunter Valley wine region near Sydney you could do in a day as well, or you could find a nice B&B for an overnight.
I'll try to think back some more - I'm sure you will have an awesome time! PM me if you have any specific questions.
Mommy to
DD1 (12/06)
DS (6/08)
DD2 (10/11)