Normal ACH (Automated Clearing House) transactions in the US tend to be free. I can move money from one bank to another bank without any fees. It usually takes anywhere from 1-4 days. This is moving US dollars. Not sure what the cheapest way to move USD to EUR is and likely not free.
This is largely dependent on your account type and bank. There are many types of accounts that will not charge for wires or will only charge on one side (incoming/outgoing) of the transaction.
I don't pay outgoing wire transfer fees on my business account.