As they say, when in Rome,do as the Romans do. Take a bus. Unless you want the luxury of a private car, board a bus from the airport to Termini station. The rent will be around 7 Euros per person.
But if you are 4 people, then it makes sense to hire a car. These are some operators: www.romecabs.com and www.romeshuttlelimousine.com. Prefer a Roman taxi with fixed rates. Or there's this private carpool rental service called blablacar.com. Heard it's good. about 5 years ago