Australian don't have national Insurance numbers.. however there are several Medicare offices in Perth, Medicares website is unsure of the suburbs let me know what area you'll be staying and I can tell you the closes to you.
If you'll be working you'll need a Tax File Number if you go to https:--iar.ato.gov.au-iarweb-default.aspx?pid=4&sid=1&outcome=1 you can read a bit about it and also apply online for a number.
As for a bank account there are many banks around and most charge lots of fees... I have heard ANZ is one of the better ones, it has an account that cost $5 per month and that allows you unlimited transactions as long as you use only ANZ ATM's.. most banks charge you to use other banks ATM's and charge you again if you make more than 5 transactions a month.. they suck I know. You should be able to set up an account after you arrive at whichever local bank you choose just remember lots of ID.
Perth is a beautiful city, I was born there and have lived there most my life.
Good luck and all the best
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