Best urban zoo? Taronga.
Best open range zoo? I've only been to Werribee, but Dubbo has the best collection and is the only open range zoo that would be a full day out. Although really it's a two day trip minimum, because it's 5 hours from Sydney. Werribee, by contrast, is 30 minutes from the Melbourne CBD.
Best native wildlife park? Healesville Sanctuary. But you'll be able to see virtually everything they have there at Taronga.
Best aquarium? Sydney Aquarium. However if you're going to the Gold Coast go to Seaworld as well. Dolphins and the best shark enclosure you'll see in Australia.
Basically, if I were travelling to Australia from overseas I'd be aiming to go to Sydney (and go to Taronga and Sydney Aquarium), Dubbo (Western Plains Zoo) and Melbourne (Melbourne Zoo, Healesville Sanctuary). Hardcore zoo fans might add Werribee to the mix, but there's really nothing there that you won't see at Dubbo, Taronga and Melbourne. If you're coming after November Adelaide Zoo will have giant pandas.
Basically anything else depends on being in the area. If you're going to North Queensland check out Cairns Wildlife Safari Reserve or Reef HQ (neither of which I've been to). If you're going to the Northern Territory there are two publicly owned wildlife parks that are reputedly excellent - Alice Springs Desert Park and Territory Wildlife Park near Darwin. Go to Australia Zoo if you're an Irwin fan, and Perth Zoo if you make it as far as WA.
If you're coming for a full zoo tour make sure you buy a membership to one of the societies. Any membership to one of the major public zoo societies (NSW, Vic, SA, WA) gets you reciprocal rights to the other states, so it *will* pay for itself.