Hi Brian,
The Sydney area doesn't have to many places like that but I do suggest you take a trip to the Sydney Wildlife World at Darling Harbour. They have all sorts of Australian creatures there and do have a daily bird show with some fantastic native birds, well worth a look, this is the web address www.sydneywildlifeworld.com.au . If you're travelling further a field let me know and I'll give you some pointers.
Gordon.
Brian,
You could also try Featherdale Wildlife Park. They are out in the suburbs but smaller than Tooronga Zoo. If you go on a weekday when the kids are at school you can enjoy yourself without the crowds. http://www.featherdale.com.au/index.htm