St George offer banking closer to what you are probably used to in the UK than most of the majors.
http://www.stgeorge.com.au/They offer two 'current' or transaction accounts,
Freedom which is a more traditional Aussie account with quite a few fees and limits, however with a balance over $1000 these fees are reduced and can be zero.
Freedom Plus which is a $6 per month fee for unlimited transactions regardless of balance.
Both accounts offer internet banking, phone banking, they are one of the few Aussie banks to offer the option of a
Visa Debit Card.
They also offer a number of different types of savings accounts, the most common one is their Power Saver (if you are in NSW,ACT,SA or NT you can link the transaction and savings account and earn extra interest, look for 'Power Plus' on the website)
In addition they offer all the usual financial extras such as home loans, credit cards (various types with varying fees and benefits) etc.
Until April they are allowing use of other banks ATMs for no fee but after that it's back to business as usual.
The only drawback is that their ATM's are fewer and further apart than the 'big 4' banks but you can still take out cash via EFTPOS when shopping etc.
My own personal opinion of them so far has been very good. They issued a proper plastic ATM/Eftpos card in the branch when I opened the account, set up my phone and internet banking, PIN etc. And so far have provided an excellent service.
The Visa Debit card arrived two days after opening the account and I was able to go into the branch and activate it/set up a PIN so it was useable as soon as I got it.