Following are the largest 4 banks in Australia: 

1. NAB (National Australia Bank)
2. ANZ Bank
3. Westpac Bank
4. CBA/CommBank (Commonwealth Bank of Australia)

 I think all of them would waive (cancel) the monthly fees if you deposit around $2K each month. In other words, if you get your salary sent to this account, you don't pay fees (for most people).

I think there are some websites, which compare bank fees and services. Search Goggles for:

Compare Bank Fees for saving accounts

CommBank is offering the following:

No monthly account fees for the first 12 months with our Everyday Smart Access account.

The $4 monthly account fee is waived for the first 12 months if you’re a moving to Australia customer who opened their account using the online application form. After 12 months the fee will apply unless you deposit at least $2,000 a month (or $1,000 if aged 21 to 24)

https://www.commbank.com.au/personal/can/moving-to-australia/setting-up-your-banking.html?ei=gsa_articles_bank-account-before-moving-to-australia

The only bank among the big four banks that provide No monthly account fee is NAB BANK.

There are some banks that don't apply monthly account fee. However, they don't have branches in all suburb

HSBC

CitiBank

Macquarie Bank

ING Direct (if you deposit 1000 each month, you can withdraw from any ATM with no Fee)

From my personal experience, I think the best mobile banking is Commonwealth Bank despite it does not support android pay nor Apple pay until now but you can use its own mobile payment if you have an android mobile with NFC feature.

I used to have a bank account with two of the big four banks in Australia, CBA, and NAB

However, I recently opened an account with UpBank and I have to say that it exceeded my expectations.

They offer an interest rate of 1.85% per annum which is one of the highest in Australia.

The mobile application is very easy to use and very innovative, for example, you can see all your incoming bills and get notified so that you don't forget to pay them. You can see transfers to a specific person as if it's a conversation between you too, so you can see the full history very easily without having to go through the hassle of filtering transactions.

You can open many savers accounts for multiple purposes and transfer money between them very easily

You can enable a feature to round you spending transactions into savings, for example, if you buy a coffee for $4.95, it will automatically transfer 5 cents to your savings account.

No fees on international transactions

No fees on international ATM use

No bank fees what so ever

and more importantly, it supports pay ID, so your transfers to the big four are instantaneous.

They also have a referral program, so if you are interested after going through the website and want to sign up, I can send you a referral link which will earn you some cash $$.

https://up.com.au/