Most of NSW is not worth visiting - except Sydney and the far North coast. Sydney is great because it's city living on the beach. Good beaches. That's something Melbourne doesn't have close by. Sydney is great if you surf. There are always heaps of things to do in Sydney too.
QLD is much nicer if you are wanting to spend all your time at the beach. Brisbane is not so exciting, but the Sunshine Coast and Surfers paradise are pretty sweet. There are a few small towns south of surfers that are awesome, with really good surf. If you can get up to the Great Barrier Reef, do it. It's an amazing place.
Canberra is ok if you are passing through, but there isn't a whole lot of stuff to do here. Most of what we have is touristy stuff like parliament house, the national gallery, the national museum, etc. If you are into that sorta stuff then Canberra is good. It's by far the most beautiful city in Australia.
Melbourne's a good city to visit. I wouldn't describe it as being as flavoursome and diverse as Mark did, but it's certainly one of the more diverse places in Aus. I'm not as pro Melbourne as some of the people on here, but I still think it's a nice place. Very different from Sydney.
Perth is... well.... kinda like the other main cities, but it's on the other side of Aus. Same goes for Adelaide. There isn't much that makes them worth visiting if you have limited time IMO. If you had a year to spend here, I'd say go and visit them, but otherwise I wouldn't.
Of course, I think you'd be insane to come all the way over here only to spend time in the cities. The thing that really sets Australia apart from the rest of the world is the crazy landscape. Try and check out Ayers Rock/Uluru, and some of the top end rainforests. Darwin is good for this.
Darwin is the sorta place that you visit for a week, and then leave and never go back. It's remote and in the middle of a really fascinating environment. It's surrounded by Kakadu and Arnhem Land which means it's a great place to go if you want to experience the Australian bush "Crocadile Dundee" style. It's pretty strange up there compared to the rest of Aus, and just about everything up there can kill you
It might be something that interests you.
Australia is pretty good compared to most places when it comes to accepting foreigners. Being an American, some Aussies will expect you to act ignorantly and like you are better than the rest of the us. As long as you don't do that, you'll be fine!
I'm too lazy to go back and change what I typed.
