Melbourne, I Would choose to visit hands down, over Sydney anyday. I am Australian, and have lived in both cities, but come from Brisbane which you might also consider. I have to warn you that in Australia, no matter where you go ( although the further north you head, there is the same stinking hot humidity that drives people crazy) will be hot. Melbourne is great for culture, chilling out, shopping and sightseeing within and around.
Brisbane, same. Plus up north to mooloolaba and Maroochydore, there are top beaches. Also south towards the eternally stuck in the 80s Gold Coast.
Sydney, do a day or two of the ferry to manly Beach, forget Bondi unless you are pretty flush with cash given all of the expensive Restos that have popped up. Head on south to Kiama and the southern highlands. Or, up north to the central coast.
Tokyo, too much to see and do to write down all. I've been there twice and even a week is not enough. From the beautiful gardens of Shinjuku to the nightlife of Ebisu plus their brewery ( if you like Japanese beer) to the shopping district of Shibuya to the tourist trap but interesting place of Asakusa to concrete jungle of Ginza , avoid Rappongi. Even Lonely Planet couldn't tell me everything I saw and did there. But also get out of Tokyo and hop on a Shikansan train to Kyoto, Nara and surrounds.