I prefer a local credit union with online facilities. I emphasize local because sometimes you just have to be able to sit down face to face with someone to resolve an issue. By local, I mean within a reasonable driving distance if you really have to go to their office.
As an example, I used to do all my banking with a credit union located hundreds of miles away in my former home state. All of a sudden, they up and changed their name and issued everyone new ATM cards with randomly generated access codes. I wanted by old code back, and they said, "Sure, just come on in." It turned out that their system does not allow their customer service folks to change an access code, other than to generate a new random code. It requires the member to insert the card in a special reader in the CU office and to enter the new code into an attached keypad blocked from the customer service person's view. I am now in the process of switching everything to a new local credit union.
I prefer a credit union because they are member owned and management is accountable to those members. Banks are far more profit driven, as recent events have shown.
I have my retirement check deposited directly into the CU account. I can also move funds between it and an investment account elsewhere electronically. I buy everything except small change items with a credit card, which I pay off each month using an ACH (Automated Clearing House) draft of my checking account.
I have never signed up for any bill paying service, and I have never paid a service fee to have a bill paid electronically, yet almost all my bills are paid that way. I write very few checks.
I allow reputable companies to use monthly automatic ACH drafts of my account to pay recurring bills. I do not do this with any bill that may have pass through charges from third parties, such as credit card companies or telephone service providers. Those I pay with a manual ACH draft which I approve each time. I have be doing this for many, many years, and I have not had a single problem with it.
I usually get cash back using my ATM card at grocers for my walking around money. My credit union does not charge for this, nor do they charge anything for ACH transfers. If they did, I would go elsewhere.
To be clear, as I understand it, bank/CU bill paying services process and pay your bills, usually for a monthly fee. ACH payments just process the bank drafts.
I also take advantage of the Co-Op inter credit union ATM network. It offers free ATM use at participating credit unions. You do have to be very careful to make sure there will no charge at either credit union and your home credit union absorbs the network fee. The Co-Op network also allows access to other networks, such as Pulse. Just expect to pay an ATM charge at the foreign institution.