Buses and underground by challenger
Tickets for buses and underground are only E0.77 and valid for 75 minutes on a one way journey. As such day or week tickets are usually not worth it: There are so many attractions in Rome that you will probably spend a lot of time walking from one attraction to the next and avail of public transport only once, maybe twice a day. Public transport is available from around 05:00 until shortly after midnight in very regular intervals.
Remember: You will need to purchase your tickets at a ticket machine or a tobacco store PRIOR to your journey. Once inside the public transport you will need to STAMP the tickets.