Sounds like a good trip! We were in Australia this past March for 2 weeks. Probably a different sort of trip than yours though...we had my 3-year-old son and almost-1-year-old daughter in tow. What sort of trip is yours? Family, business, other?

We arrived in Melbourne and spent one night in the city then headed along the south coast (Great Ocean Drive) for three days/two nights and back to Melbourne for 2 more nights. Then we flew to Cairns and spent 5 nights in the Port Douglas area (about 1 hour drive north of Cairns). After that we went to Sydney for 4 more nights. We enjoyed all three different areas of Australia and have no regrets. I would have liked to get to other areas as well (Uluru, Gold Coast, etc) but it is a lot of ground to cover and I am pleased with what we were able to do.

Melbourne is a beautiful city with some interesting architecture, nice coffee culture and an excellent market (Queen Victoria Market). Public transportation (above-ground rail) in the downtown core is free too which is a nice bonus. Make sure you check out both the day and night market as they are totally different vibe (runs as a night market a couple days a week I think...we went Wednesday night). The south coast is of course famous for the Great Ocean Road and I think it is worth seeing (even if you don't do it all). Some iconic (even if a bit cliché) shots of the 12 apostles and other rock formations are a must. The surfing mecca of Bells Beach and Torquay are on the south coast at the start of the Great Ocean Road as well. Additionally there is supposed to be some great wildlife along the dirve and particularly in the Otway National Park, but we didn't see much ourselves.

The area around Port Douglas was great as well. We opted for the resort-ish town of Port Douglas instead of Carins as we thought it would be a bit more laid back for the kids. The beaches are great, but we were there in stinger (jellyfish) season so didn't swim on the beach there much. Also need to watch out for salt-water crocs. Definitely worth a boat trip out to the Great Barrier Reef for some swimming/snorkeling/diving. There are lots of great places to explore in this area: Mossman Gorge, Daintree National Park, Kuranda, etc. Lots of wildlife here too (we saw wild kangaroos here) and plenty of wildlife parks where you can see native wildlife (snakes, crocs, birds, kangaroos, etc) - nice for kids as you can often interact with the animals (ie feed kangaroos).

Sydney is also a beautiful city, but I'm sure google can tell you all about that... Bondi beach was a nice visit for an afternoon a short trip from downtown. The harbor is nice for boatrides and lots of iconic pictures can be taken here (opera house, Sydney bridge, etc).

I can give you specifics for what areas we stayed and specific activities or sights if you're interested (I love talking travel...). Or post pictures of places if that helps give you an idea of whether you want to go somewhere we've been. I have some pictures up on my flickr ( https://www.flickr.com/photos/nfld_stephen/) if you are are interested.

Have a great trip!
Stephen