It really depends on how you deal with heat. If you love it, come in the summer, otherwise the middle of spring is best as winter tends to be a bit too wet in most parts.
Queensland is great for the tropical climate, Cairns or the Gold Coast are good there.
The allure of Sydney escapes me, but it is Australia's most populous city, so there must be something there other than the Harbour Bridge and the Opera House.
Melbourne is a temperamental city. It can be hot one day and then a cool change will come through and the temperature will drop 10 degrees in the space of an hour. It can go the other way too, as today's weather is proving. Melbourne's good for sitting around in a cafe, watching people and drinking coffee.
Perth has very stable, fine weather with warm days tempered by the Fremantle Doctor, an easterly wind which blows through late in the afternoon providing beautiful, mild nights. Being so very far away from anything else, Perth is a very laid-back city.
Tasmania is quite picturesque, with lovely countryside, but if you want something a little more spectacular (if a little bleak), go to Alice Springs where a visit to Uluru (Ayers Rock) is essential.