Most banks will charge you a monthly fee to use their internet and telephone banking service. You will also be charged for all transactions, though often the charges will be less than ATM or bankteller charges. Remember also that you will save money on transport to use the ATM and/or pay bills. You will also save time, and perhaps avoid late fees on bills that you previously weren't able to pay on time.
Look around at the different banks and see what their charges are like. If another bank offers better internet charges, see if their account charges are also lower. If they are, consider switching your accounts to the new bank.
You can save money on bank charges by only checking statements once a week or even less often, and by making sure there is enough money in your account to pay bills. Returned debit orders will cost you money.