MSPeconomist

The Steve Irwin zoo is a couple hours from Brisbane. Fly. It's either Gold Coast or Sunshine Coast but I can never remember which is which.

Otherwise, if this isn't a high priority, I'd suggest the following to maximize Australian animals:

Skip NZ as you don't have enough time.

In Brisbane, take the boat trip (combined ticket) on the river to see the koalas. Be sure to walk over the bridge from downtown where you look down and see sharks in the water. There's an interesting tourist place to see different types of Australian sheep (with dog herding demo shows) and a variety of other animals, IIRC accessible easily by light rail and then a walk. It's a good place to feed (friendly hand raised) kangaroos and hold koalas too. The botanical gardens near the river sometimes have wildlife.

In Sydney, be sure to take the boat to the zoo (and ride the cable car to the top when you arrive, then work your way down the hill and back to the ferry--look for big birds such as pelicans and ostriches/emus walking around loose), take the (different) boat to Manley and see its small and somewhat quaint aquarium, ideally at a time when there are divers in with the sharks. Also go to the big Sydney aquarium; the place there where you walk through a glass "tunnel" surrounded by lots and lots of sharks (that followed me) is impressive. Sometimes you see wildlife in the botanical gardens.

In Melbourne, go the the zoo (my favorite), ideally at a quiet time so you can get close to the kangaroos in the walk through exhibit. There's also an aquarium and an interesting historical prison and market. Take the streetcar to the old beach resort town on the sea in a suburb of Melbourne. I also liked the penguin parade, but it's only practical to do it as a tour. I found one that left Melbourne in the afternoon and first stopped at a small private zoo or wildlife place where you could handle some Aussie animals. Then we went to a place with lots of birds nesting (this may be seasonal) and finally to the penguin parade. It was somewhat of a "hippie" tour run out of a backpacker hotel almost across from Victoria Market, so the guide and group were casual and fun, with an interesting mix of people (but nothing that would be inappropriate for children). Other tours to penguin parade do different schedules and itineraries, but penguin parade is after sunset, so expect to return to your hotel very late. If there's time, try to do at least part of the Great Ocean Road.