If you have a chance while you are near Sydney, you might want to check out the Blue Mountains. My trip there was awesome - went hiking, saw gorgeous waterfalls, tried rapelling (they call it abseiling). The name comes from the blue tinge of the sky (you really can see it!) resulting from the eucalyptus trees in the valleys of the mountains.
Also, if you are interested in wine at all, check out one of Australia's many wine regions - they are scattered about the country, but near Sydney is the Hunter Valley. You can do wine tasting, get a tour to see how it is made, and the areas are really beautiful just to look at. Also, my drive was full of wild kangaroos! (It is an easy day trip from the city, as are the Blue Mountains, though they are a little further.)
Also, if you find that you are short on time while there or suddenly decide that you want to stay longer at one place than you expected, I would recommend that Canberra be the first place to skip. I know it is the capital, but there is very little there and it is a good hike from Sydney. There were a couple neat museums, however, and the drive past all the embassies was exciting (all of the different buildings are designed in the typical architecture and style of their respective countries).
While in Cairns, another place I would recommend is the Kuranda rainforest. You can take this scenic railcar through the area, go on the skyrail over the canopy, and hike guided or alone. Also nearby is Tjapukai, this Aboriginal park where you can see dance performances, practice and learn to throw a boomerang, try spear-throwing, etc. It was a pretty fun place.
I know that's a lot of stuff to throw out there, but I figured more is probably better. I'm so excited for you. I spent six months in Sydney and did a little travelling around the rest of the country as well - it was amazing!