Metro, Buses and Trams are operated by ATM. A single ticket is valid for 90 minutes from the first time you use it and allows you to make one journey on the metro plus one or more journeys on Buses and/or Trams. As in London, you need to validate your ticket on entry and on exit for the metro, on entry only for buses and trams.
There is also an Oyster-like NFC card ("Ricaricami") but paper tickets are the same price so the easiest thing is to buy a few single tickets, also called "Urban" tickets. There is a zone system but what is called the "Urban" (central) zone is pretty big and you probably won't go past its limits.
You can also buy a 24-hour travelcard ("Biglietto giornaliero") which is the price of 3 single tickets. This is still a paper ticket you can buy from any ticket machine in any metro station.
A more recent addition are the S-lines (also known as "Passante"), an Overground-like rail service, but running mostly underground. They connect a few areas with no metro station and you can use a standard ATM ticket.