We opened accounts online with ANZ, a cheque account and a savings account. They sent back the account numbers within a week as they have office in UK. At that point we didn't even have an address in Oz.
We could pay money in straight away which i did by money transfer using HiFX (great company) but we couldn't draw any money until we had validated the accounts at one of the braches in Oz by showing our ID giving them your address in Oz. We did this at Melbourne airport when we arrived as they have a branch in the departures area of the international terminal and drew money from the account straight away. By this point we had an Oz address as my company was providing an apartment for a month.
Within a couple of months I also applied for a credit card with them and got without any hassle, didn't even ask for any sort of references or previous credit history. Could probably of got the credit card staright away!
All in all I think the banks are all pretty much the same, you've just got to be aware that most charge you fees on your current account like they used to in UK many many years ago. Some charge per transaction, some a flat rate, some charge every time you use an ATM. We went for ANZ because they seem to have the most ATMs around and the account we have charges a flat rate of $5 per month. Though there seem to be moves for the banks to drop all the fees as UK banks did.
Other thing to be wary of is that you can't use any ATM to draw from any bank like UK now, I used a NAB ATM to draw from my ANZ account and got charged $1.50 on $100. I know there are some of the banks that work together and don't charge to use other ATMs but am not sure which.
hope this helps
Rob