Look into fees and structure of accounts at whichever institutions you are considering, from the angle of what fits your situation. Also consider whether you prefer a bank with a physical branch (in which case, your short list is whichever banks are convenient to where you live) or whether online banking is sufficient (some banks are online only). Some banks also limit how much money they'll guarantee; so if your accounts might go over $249k you should make sure your chosen bank is good with that.

Over the years I have had accounts with: Westpac, ING Direct, Teacher's Mutual Bank (formerly a credit union), NAB, ME Bank, Up, and a couple of others I don't remember. None of them were especially bad, but they did different things for me at different points in my life.

When my parents (who are not citizens, have never worked in this country, and are here for a couple of months at a time) wanted a bank account to use while here, I set them up with NAB. That was the option with a physical branch nearby that wasn't going to hit them with monthly fees for the privilege of having an account, when the account was not in regular use. (NAB has since closed the branch near us, so idk how much longer they're going to have our business.)

Final thought: there is no contract with your bank unless you take out a loan or term deposit. Pick one to start, and if it's not a fit you can move your money elsewhere. Don't stress.