There's more to being polite in Italian than you might think so let's discuss the first words out of your mouth in every situation:
Grazie
This word never goes out of style in any language when you want to thank somebody or express gratitude. In Italian it’s also used to complete your phrase politely, both in accepting or refusing an offer:
Yes, please. Sì, grazie
No, thank you. No, grazie
Prego
It's a versatile word that can do magic. You can use it in Italian for a variety of situations where spoken English has many different ways of expressing the same sentiment.
And be thankful that Italian has something so simple and so easy to pronounce in place of the myriad expressions that English, Antipodeans and Americans use.
Reactive use
In all the situations where someone has thanked you or acknowledged your help and you need to respond:
You’re welcome! Don’t mention it! Prego!
Not at all! Happy to help! Prego!
Or when you are allowing somebody to do something for which your permission was required:
Sure. No problem. No worries. Prego
Anytime dude! Prego!
It’s also used by itself with a questioning tone of voice when you need someone to repeat something you didn’t hear properly:
Could you repeat that please? Prego?
Pro-active use
In all the situations where you are deferring to someone by giving way or allowing them to go first or signifying that it is their turn to be served in a shop:
After you! Go ahead! You first! Prego!
What can I do for you? Prego!
Per favore (a linguistic false friend)
It’s not as common to use this in Italian as it is to say please in English. In fact it is rarely heard.
Polite use
When you want to make a request more gentle, you don’t need to use it to replace the English ‘please’, despite what your dictionary says:
I’ll have a coffee, please. Mi fa un caffè? or even Un caffè
Can you give me two plastic bags, please? Mi dà due sacchetti?
Could you pass the sugar, please? Mi passi lo zucchero?
As Italian is a musical language, kindness is rarely expressed by the use of special words, instead it’s mainly a question of the intonation of your voice. All you need to do is to use a gentle tone of voice, nothing else.
Therefore, if you say “Un caffè per favore” you will immediately sound like a tourist, in which case be prepared to answer the question “Da dove vieni?” i.e. “Where do you come from?”
Emphatic use
Here it is used to stress a request when you’re asking (almost begging) for something really important:
Please, don’t make me learn any more Italian! Per favore, basta con l’italiano!
~Cheat Sheet~
Grazie goes with ‘yes’ as well as ‘no’. In accepting or refusing something politely you will always use it e.g. “Sì, grazie” , “No, grazie”
Prego has many meanings, it’s often used alone either as an exclamation or a question e.g. “Prego!”, “Prego?”
Per favore bear in mind that it’s not a colloquial translation of ‘please’