There's a multitude of unique wildlife in Australia...but a lot of the weird and wonderful species are spread from one end of the country to the other and may not seem quite as 'accessible' as what you'd find in game parks over your way.
The late Steve Irwin's Australia Zoo is about an hour north of Brisbane, and is home to many of our native fauna, but if you want to find platypus, echidna's, Tasmanian Devils in the wild, you'll need to do a bit of traveling.
If you're in Brisbane make sure you take the time to head to central and northern Queensland - spending a week or so visiting the Whitsunday coast and tropical far north make for a excellent road trip. Even more so if you're into SCUBA diving/sailing/fishing etc.
Fraser Island (~3 hours north of Brisbane) is also a great photography destination - not only for the flora/fauna but for the general environment.
Or you can head 1 hour south to Surfer's Paradise on the Gold Coast, where some really interesting creatures hang out (hint; they walk on 2 legs and hang around in nightclubs!) 
Have fun and I hope you enjoy your stay down under!