To open a bank account in Madrid you may need your foreigner’s identity number or numero de identidad de extranjer@ (NIE). This will either be included in the temporary visa in your passport or you will have to wait to get your visa card (tarjeta de identidad de extranjer@ or TIE) to get the number. Some bank branches will need you to have the number and card to open the account whereas others will be fine with you using your passport number, and coming in later once you finally have your identity card. To open an account you will almost definitely need to use Spanish, and remember you can generally bring a friend with you to help.
I opened an account with BBVA, using my NIE which was included in my temporary passport visa, and just had to come in (to the same branch) once I had my physical foreigners card too. The branch you use to open the account will be considered your main branch, and you will need to return to the same one to take care of logistics. I chose a branch out in one of the little pueblos that I worked at, which had the advantage of almost never having a line of people waiting, but the disadvantage of being far away and the fact that I only went to that town twice a week.
I have heard great things about the bank EVO. They are small and don’t have many branches, but you can take out money from any bank’s ATM (cajero) without having to pay a fee.