Taronga Zoo is a must - even if you don't like animals. The view is spectacular, as is climbing the harbour bridge (if you have the money and fortitude).
Take a picnic to the Botanical Gardens, walk around McArthurs Point. A ride on the old green and yellow ferries is a must and a great way to take in the harbour.
The Blue Mountains are lovely to visit with some very quaint bed and breakfasts, go see the Three Sisters and there's a pretty steep cable car ride. Great bushwalking. Much more peaceful adventure than Sydney.
Brisbane is more like a big sprawling country town than a city. For shopping and just hanging out with the locals, go to Brisbane Mall - large music centre called Allan's Music is definitely worth a visit.
Walk across the bridge and head for South Bank where they held World Expo in 1988. There's still a man made beach, volleyball, picnic areas, tons of restaurants and cafes, small galeries.
Then there's the Lyric Theatre that always has Opera, Musicals or other large music events. Pretty much opposite is the Conservatorium that always has performances. Across the road is the ART Gallery.
You can catch the ferries to back on the other side and try your hand in the Casino (also a good place to get some cheap food!).
Markets are held on a Saturday, and if you're lucky to score a particular Sunday, there's free classical music sometimes at the Botanical Gardens, sometimes at South Bank.
The Gold Coast is South of Brisbane with some very good beaches, heaps of night life and food! and of course there's the theme parks: Movie World, Dream World, Wet 'n' Wild and Sea World. You can get a three day pass that includes three of them, and sometimes includes accomodation as well. If you're rich and money is no object, definitely stay at NARA RESORT which is attached to Sea World.
Topic: Tourists in Australia