The store where Italians drink coffee is called a bar. The sign outside one of those establishments may carry additional specifications characterizing it. For example, it could be a bar tabacchi, meaning that it sells cigarettes and stamps. Or a bar pasticceria, meaning that it sells fresh pastries. Or a bar gelateria, which sells gelato. In a bar you can drink coffee beverages, fruit juice, alcoholic beverages and also eat pastries, panini, gelato, etc. (with many possible variations).
Andare al bar a bere un caffè is to go to a coffee shop to drink an espresso. We also say andare al bar a prendere un caffè (to get an espresso). It is a well-known fact that Italians drink espresso and that Italian espresso is excellent. Note that we don't use the word espresso, but if you do, people usually understand what you mean. I have previously written posts that detail the different kinds of coffee and milk beverages that is possible to order in Italy: here, here and here.
While in Italy, you will not be able to replicate your experience of ordering and consuming a coffee beverage in an American coffee shop. In Italy, each beverage comes in a predefined size (an espresso cup or a cappuccino cup) and with a standard type of milk (usually pasteurized whole milk). Concepts like medium, large and non-fat are foreign to us. Decaf is (barely) accepted. You may be able to obtain soy milk in some establishments.
One important aspect of the transaction is the payment and placing of the order. In the U.S., you pay at the cash register and order the beverage of choice, then your order gets transmitted, possibly by computer, to the person in charge of making your drink. After obtaining the drink, you can walk away or sit at a table and consume it on the premises.
In Italy things are not so straightforward, mostly because there are no set rules that everybody follows and there is no computer next to the coffee machine. In most places, and always in the busy ones, you pay at the cash register and obtain the receipt. You then show it to the barista and repeat your order. He or she will usually tear the receipt a bit and give it back to you: do not throw it away. You are then given your beverage and you are supposed to consume it while standing at the counter.
Detail number 1: you are required to keep the receipt for any purchase you made until you are within some distance from the store (bar, restaurant, grocery store, department store, etc.). Outside the store you could be stopped by a policeman, in which case you should show the receipt. Failure to do so could result in a fine.
Detail number 2: you will pay extra if you sit down at a table. If you want to take your time drinking your espresso and maybe have a chat with your companion, you signal it by sitting at a table. A waiter will take your order and deliver it and you will pay before leaving, either at the cash register or to the waiter. Whatever you order, it will cost you more than if you had ordered it at the counter.
Sometimes the barista is also the cashier, in which case he or she will both get your money and prepare your drink. Other times you can order and consume first and then pay before leaving. The variability of arrangements is wide, so the best thing I can do is to alert you of what is possible. If you try to do things in a different order, you will be gently reminded of the correct one, no harm done. So, if you order your beverage before obtaining the receipt, the barista will ask you for it and allow you to obtain it without bad feelings. In some places, a sign asks customer to pay first and consume later.
A few more notes: in Italy we don't walk around drinking coffee, so the concept of 'to go' is quite foreign. Because we do not routinely use disposable coffee cups, we don't see the need to bring our own to reduce waste. We also don't feel compelled to fill containers to the brim. In fact, in some cases, that is a veritable no-no. For example, a shot of espresso will never fill the cup.
In Italy we drink un caffè for the pleasure of tasting an intensely aromatic nectar that leaves behind a heavenly aftertaste. Water makes us suspicious, which is the reason why, when you order tea, you get a minuscule amount of water and no refill. (This is a very personal opinion, which I will hold on to until I find a better explanation.)
One of the things I do soon after arriving in Italy is bere un cappuccino al bar. This time I drank my first one at the Bar Cavour, an old (historical) establishment in Bergamo, a city not far from Milan, which is not known as much as it deserves to be and which I have always loved. In this case, the barista doubled as cashier and the bar was not busy, so we got our beverages and paid for them afterwards.