There's the most kickass Bill Pay system ever and DEFT, which allows paying by several methods including the credit card, potentially gaining reward points for paying rent. (Reward credit cards may be pricey though.)
Overseas wire cost a measly flat AU$20 a few years ago in WestPac, local transfer was free and quick to a low-risk recipient (and most recipients are reasonable enough to go on with a transaction if you send them a PDF proving you sent the money). If you run a business, your company account can be accessed via the same customer ID.
Suspicious transactions are spotted and the bank calls you if they suspect foul play; the false alarms are rare. Fraudulent transactions (at least minor) are reversed - happened to me once and to my wife several times. No US-like secrecy for bank account numbers, you normally publish your bank account for the people to make transfer to. Tokens are used for business accounts, while the consumers don't have to bother with these.
There's generally no banking fee unless the account is dormant, and then it's about $5 or something (not $20 - $40 like in the US or Singapore).
Nobody uses cheques, obviously (although they do exist).
In terms of foreign currency transfer, there is also OFX, a nifty service where you can keep your incoming funds until the exchange is more favourable (nope, Transferwise doesn't have it), and that charges you the fraction of what the banks take.