Hi Beccy,
My partner and I moved to Adelaide last November and love it here. It's such a friendly and welcoming place and affordable to live here.
I'm a teacher too and can give you advice on the process. I kind of figured it out on my own but wish the process had been more streamlined.
Firstly, yes you do need to go through the same rigmarole as the AITSL with the Teacher Registration Board of SA. You need a different police check this time. Details at: http://www.trb.sa.edu.au/fitness-propriety .You need to fill another form in with the UK police (same place as you had to apply for the Visa Migration police check) called an Overseas Criminal History check. This is like the DBS/old CRB certificate in the UK. Request one asap and the TRB will email you a letter to print and post with your application to the UK police as you can't get the check without the letter.
You also need to do a Mandatory Training (full day/7 hour) course. It's basically Child Protection training. You can book them online. The trb has a list of recommended providers for this training too.
To get Teacher Registration you will need the Police certificate, Mandatory Training Certificate, your AITSL certificate plus ID documentation that you already used for AITSL and the visa (stupid to duplicate, but they seem to do that a lot here!) signed off by a JP. I met one on our first day here and he signed them all off for free. They don't charge here for that, it's more a position of professionalism and respect (like teachers are able to witness passport applications in the UK).Thereare lots of JP's so just ask around when you move here.
Anyway, after you have all the docs together and signed off, I'd advise physically walking into the offices in the city and submitting the application. Someone goes through it there with you to check it's all in order. Plus there's no risk of losing documents in the post. I got my provisional registration certificate back in less than two weeks (and that included the christmas period!) You will get a provisional registration too. After one year of service you can then then top up to the full registration.
Hope this helps. Feel free to PM me if you have any other teacher queries.
Good luck with the move, it's pretty cold here at the moment but it's a lovely place!
Emma :-)