Info on Sydney Australia

I won a trip for my boyfriend and I to go to Sydney, Australia. I still need to start booking it with the travel planner but I am wondering what type of things I should definitely do while there. We are getting free passes for the Syndey Harbour Bridge which I hear is really awesome and I have been looking online and see a bunch of zoos, aquariums and sanctuaries. For those that live there/have been there, what do you recommend doing?? Also, I wanna go when the weather is decent and it looks like September and October are good months. Is that correct? Thanks!