Hi
When I emigrated out here in 2003 I went and got a special licence to fly on. As previous posts say, rock on up to any CASA office with medical, log book and UK licence. You will also be provided with an ARN number. CASA will also give you a form (if memory serves me correct) for you to take away, fill out and sent to CAA in UK, allowing them to release any info they have on you (basically verifying your licence is legit). They will not issue the special licence until the CAA have given them this info, which in my case was within a week.
Also not entirely sure about ASIC situations for overseas pilots, but as far as I am aware "every" person needing airside access at airfields with RPT traffic must have a current ASIC card. Now given that some people have been waiting months for these to be proccessed, you may find yourself flying into a lot of small uncontrolled aerodromes until you get your card.
Still saying all that, it great place to fly with some fantastic scenery! And cheap. Currently flying a new 172 for cheaper than what I rented a 30 year old 150 for in the UK!
One last tip also, practice your navigation skills, cause its pretty barren as soon as you head inland!
Good luck with it all!