2 of the best free bank accounts at the moment are from the HSBC and NAB, there are a few others out there but they have terms such as you must deposit at least $2000 a month. The HSBC and NAB ones offer proper free monthly accounts and are good starts into saving money on banking:

HSBC - Day to Day Bank Account:
No Monthly Account Fee Banking - it is possible | HSBC Australia

NAB - No monthly account fees ever:
No monthly account fees ever - NAB

Bankwest - Rewards Transaction Account -

No Monthly fee bank account if you deposit $2000 is worth looking at if you buy from overseas or travel abroad a lot the
Bank account that's great for overseas travel & online shopping! Rewards transaction account

This offers a rewards on money in the account, however it's pretty rubbish reward and that's not the reason to use it, move your money to instant access cash savings. You need to have about $6000 in the account to earn enough points for a PS3 dual shock controller, a value of about $50, where as $6000 in the instant access cash savings with an interest rate of 4.75% will get you $285, almost 6 times the equivalent value of having the money in the account.