There is an ATM near a Tribunal metro entrance, near the piece of real estate that used to be my favourite bar in the city.
The ATM belongs to La Caixa, I’m pretty sure, a Catalan bank.
One night, in need of cash for bar hoping, I used the ATM, and pleasantly found there were language options. I thought I chose English, but the screen was off a little and I actually chose Euskera.
Basque, I had chosen BASQUE.
Basque is believed to be the oldest language in Europe, with no known linguistic relatives. The language is believed to have ties back to the Neolithic era, and perhaps even before that. An ancient, wild language, with crazy letter combinations and no similarity to my beloved Spanish.

And I had to make a bank transaction in it.
Thankfully, as anyone who lives in modernity knows, bank ATMs are pretty standard, and the steps you go through to withdraw money are pretty similar regardless of bank. I got my money despite the disorienting use of unfamiliar accent marks common in Basque.

This ridiculous, pointless story is but to highlight the need of cash in Madrid and what a foreigner might need do to obtain it.
Again, the ATMs are essentially the same as in the US. There were a few blessed machines I came across that gave out various denominations of bills and it was GREAT. I mean, you could get 5€ or 20€, unlike infuriating ATMs in the states where you can only withdraw in amounts of $20s. This was pretty rad when my bank account ran low.
Almost no bodegas accept debit/credit cards, and the ones that do begrudgingly accept them (often with a minimum so high you forgo whatever you thought you were about to buy).
Coins are as common, if not more so, than bills. I sort of (really, really) loved this. I felt like some kind of medieval knight being paid in gold (shhhhh they were just Euros shhhh I don’t care its cool). Obviously you can’t get these from an ATM, but most everywhere you pay with cash will give you back change in coins– so everywhere.

Get yourself a sturdy wallet with a coin pocket or a coin purse before you head to Europe, and it’ll smooth the transition, promise.