Melbourne is cool city for sure, but depends what you are looking for. If you are looking for good food, theater, museums, nightlife, etc. then Melbourne makes sense. If not, I'd skip. I was there in July and it was pretty cold. I don't recall anything particularly remarkable about Melbourne outside of typical city stuff. They are pretty serious about their Aussie football there. Closest equivalent city I can think of vs. North America is Toronto or Montreal.
Would you pick Sidney over Melbourne?
Yes. Sidney is like taking the best of San Diego (weather, beach) and San Francisco (natural beauty, city amenities). Great city.
Agreed. My daughter (no pics) lived in Sydney for a year after college, and I thoroughly enjoyed my two week visit there.
A trip to Featherdale (
http://www.featherdale.com.au/ ) is great to get up close to koalas, kangaroos, and wallabies (and other animals as well). You can get there via public transportation, but it takes a couple of transfers. A neat way to see the animals, though.
Another popular trip is a day trip to the Blue Mountains just NW of the Sydney suburbs (images:
https://www.google.com/search?q=blue+mountains&biw=1600&bih=751&tbm=isch&tbo=u&source=univ&sa=X&ei=0r5XVIK0OYaqyATJmIHwAw&ved=0CE0QsAQ ) The small group guides are very entertaining.
If the weather is warm, your kids would enjoy a trip to Bondi Beach or, maybe better, a ferry ride across the harbor and over to Manly Beach. The ferry ride is a great way to see the harbor, the bridge, and the opera house. If you all enjoy a hike, there's a
fantastic coastal walk along the cliffs and beaches starting at Bondi Beach and heading south for several miles (you could walk one way and catch a bus back into the downtown area).
Traveling south, the coast includes the twelve apostles (images:
https://www.google.com/search?q=australia+twelve+apostles&biw=1600&bih=751&source=lnms&tbm=isch&sa=X&ei=tb9XVLTDEczpoASivoL4AQ&sqi=2&ved=0CAcQ_AUoAg )