I came over last year on a sponsorship visa and am now going through the permanent residency process.
You can apply for any job once you qualify, you just need to get your employer to sponsor you, the whole process took me 3 months from application to gain sponsorship. Or you can simply just apply for full residency once you qualify.
Once you qualify you have to get the APODC to recognise your qualifications so you can practise in Australia.
Information from sites below: