To buy stamps, you go to a tobacconist of course!
Let me explain.
Poste Italiane doubles up as a bank. Due to the many people who use the post office’s banking services, and a somewhat ‘relaxed’ speed of service, this means that post offices often build up very long lines; to get round this the sale of basic national and international stamps has also been outsourced to licensed tobacconists. This applies to stamps for letters and postcards only; to send a parcel you still need to go to an ufficio postale (post office).
If you do have to go to a post office, when you enter there is usually a machine by the door that lists a confusing choice of services, each with its own button:
In the case of posting a letter or a postcard this will be Servizi Corrispondenza e Pacchi (or in some cases Filatelia, though this is usually for stamp collectors).
Having pushed the right button, the machine will spit out a slip of paper for you which bears a number for you to wait your turn on, preceded by a letter based on your requirement. The speed of letter/number tick-over on the board isn’t obvious but is apparently controlled by computer and based on frequency of use and availability of staff.
Due to the times involved before you get served, there are always seats in the waiting area, so sit down, relax, and get out a good book to read – but don’t forget to keep checking the overhead screen for your number, which, when it finally comes up, will have a sportello number that indicates which to which window you need to go. If you miss your turn you will have to start again! So don’t drift off while you wait.
Italian for stamp is francobollo.
Tip: try to visit a small branch away from crowds where you can sometimes get served with no wait at all; in larger post offices though, and at busy times, you could be waiting for more than an hour.
For a slightly more speedy service, the Vatican post office (of which there is one branch within the museums, one outside the museums to the right of St Peter’s Square, and a mobile one in a truck that is often actually parked on St. Peter’s Square, to the left as you look at the basilica) is much quicker in terms of fulfilling its basic services. According to rumor, when you send a letter in the Vatican Post Office, it uses a courier as far as the Swiss border, and then your mail is deposited into the Swiss postal system which is much more efficient and faster for international mail than the Italian post office.
Note that you cannot use Italian stamps at the Vatican and vice versa – they’re different countries!
Final bit of advice: don’t buy those prepaid tourist postcards with private ‘courier’ stamps on them. Some of them are a scam and never get sent, while some of them take months to arrive.
Or you could send an email. Everyone’s on email these days: you simply have to be.