If you want to experience...

 
Sports you should go and visit Melbourne, where each January they hold an international tennis tournament, that they call Australian Open. Australian Open is held in Rod Laver Arena, named after a successful tennis-player by name Rod Laver.
 
Nature we recommend that you visit Northern Territory and Victoria. You get great days filled with adventure. You go on safari and lots of other fun things.
 
Shopping we highly recommend Sydney, where they sell everything from second-hand and expensive designer-clothes. At Oxford Street you can find Sydney's best shopping-area. Mid City Center, Queen Victoria Building and Strand Acarde is within walking-distance and offer beautiful shopping-experiences.
 
History in Australia there's Australian History Museum in Sydney, where you can look at over 3500 subjects that reflects to Australian history from early colonization to present day.
 
Coasts, the right place to go is Queensland. There's many beautiful coasts like Sunshine Coast, Brisbane and Cairns.
 
Adventures, there's a lot to do in Australia. Many adventureres finds their adventures in the citites Adelaide, Brisbane, Cairns, Darwin, Melbourne, Perth, Sydney and Monkey Mia. Outside of Melbourne, Adelaide, Perth and Monkey Mia you can swim with dolphins. You can also snorkel in Queensland, for example, in Great Barrier Reef, the world's largest system of coral reefs. Sailing by Whitsundays Islands and hikes in national-parks like Kings Canyon, Ayers Rock and Kata Tjuta.