I am a teacher in Australia, and I can provide my humble opinion! I think it is fairly easy to apply and enroll in the Education Department, but the difficulty is in finding a full-time teaching position. In most of the city areas, it is very difficult to get a teaching position. Howver, I think there is much more demand for teachers in the country, if you were willing to give that a go. It is also an advantage to you if you are a catholic or christian, as you will have more chance in getting a teaching position in a christian or catholic school.
I'm not sure which specialisation areas are in demand, perhaps special education http://www.teachinginsa.sa.edu.au/
might help you with more information about teaching in South Australia (I'm not sure of websites in other states). I think the website has open teaching jobs on it, which may help you see which specialisation areas are most in demand.
Also, have you ever heard of the teacher exchange program? I don't know much about it, but basically if you are a teacher in America, you could sign up for an exchange with a teacher in Australia to swap classes for 1 year. It might be a good way to experience teaching in Australia and just get your foot in the door,