So I took a very similar trip to Australia 2 years ago, nothing was really planned outside of the opera house and reef.
Bondi beach was really cool, we did a lot of climbing around rocks and picture taking there, but the main attraction is the house in Sydney. I heard they completely renovated a lot of the Darling Harbor, so I'm sure that's really cool now, plus there are some aquariums and things around there. The tower there (can't remember name) is neat, you have a couple options of either going out and being strapped on, or just staying inside and snapping pictures. A famous church is near as well, my friend got us kicked out because he wouldn't stop taking pictures, but whether you get some or not, it's a neat place to see regardless.
Cairns was without a doubt the best part, we did the reef and some trip up into the hills for Koala bears and reptiles, so for me it would be like living near Venice, CA just better. There are a lot of friendly tourist pit stops that will help you plan things out, and they will also group things together for you for discounts and using their service. We just snorkled because you see everything as you do with diving, except it's safer and you have no restrictions on flying afterwards. Tons of cool shops and things to do, you won't want to leave I promise.
Melbourne for me sucked, I don't drink, and there's basically a Casino and lots of pubs to jump around to. We ended up renting a car and driving 2 hours out to some beaches and the wall which was awesome, but the city itself wasn't really my thing, however it's probably right up most peoples interest. We checked out the Dracula's Cabaret (no nudity) and it was a blast, it was honestly probably one of the funniest and well put together broadway type things I've ever been to. Then there is also the Penguin Parade as well, they only come out at night, so if that's an interest it's not a day time thing.