If you go to Rome, please check your travel from the airport.
While you can pay about 50€ for a taxi or buy the 18€ ticket for the express train, I suggest to get a regular train to termini. That one is only 8€. Follow the signs to train station at the airport and buy a ticket in the machines right there.
In the city, but a week ticket for public transportation. You can certainly use it for all subways and busses in the city until Sunday.