Can we transfer money to or from an account in another European country for free?

You will have to pay a fee every time you transfer money to another country, even within the EU. You can set up a standing transfer order with your bank - say if you are making monthly mortgage payments to a foreign bank on a property you own in another country - but there is going to be a charge on every transfer

You need to check with your bank to see what electronic bill paying services they offer and what the fees are. It may be a monthly fee, or per-payment fee. It depends, too, on what kind of account(s) you have. For example, some premium checking account services will include some of these things at no additional charge than your regular monthly fee for the account. If you have, for example, a mortgage or car loan with the bank where you have an account, they will typically offer automatic payment without any fees. On international transfers, there will certainly be charges, including foreign exchange fees.

Have you gone to your banks' websites and looked up their services and fee schedules? That's how you start.

uh..you are moving money. There will be a fee and there will be a paper trail.

Wire between banks is the simplest.

why would any bank do that for free?