Italian bars normally serve soft drinks, coffee and snacks as well as alcoholic drinks. People tend to mill around the counter drinking a quick coffee standing up, and often there aren't many tables to sit at. In big cities and places where there are always a lot of people around, you won't get served unless you first pay at the cash desk, you'll get a receipt, uno scontrino, which you then take to the counter, give to the bartender and tell him your order.
You might prefer to sit down and be served by a waiter or waitress, in which case you pay when you've finished.
As you wander the streets of Italy you are likely to see "torno subito" scribbled on a sign in a shop window, this means "I'll be right back". However, you may find that what this really means is "torno prima o poi", which is "I'll be back sooner or later"!