Taxis can be slightly expensive in Rome, you will probably pay anywhere from 30 to 50 Euro depending on the airport to get to the city. For our recent 3 Day Visit Touring Rome, we did a lot of research and picked the below option. There are lot of people who will ask you for transportation and you will be tempted to pay as tourist to avoid hassle, but all I can tell you is that you should try the option as it worked out well for us.
Direct Bus – SIT bus and Other Buses :
We landed in Leonardo Da Vinci Airport and took the SIT Bus shuttle. It costs about 6 Euro for one way and 9 for round trip. Check fares here : http://www.sitbusshuttle.com/en/timetable-and-fares/ . Depending on where you want to go, you have about 3 bus operators in the airport who will take you to the other parts of the city. You do not need to take any reservations for these buses as you can pay them at the counter in the airport itself. There is no discount for buying before. In the Fiumicino airport, the buses are at the entrance of Terminal 3, there are specific bus bays. You can go there and buy tickets.
Once you land in the parts, then you can take the local bus. Trick is to buy the bus tickets or pass like 48 hr or 72 hrs. You can only get them at ticket vending machines or book stalls, etc. So, you can pick them up at the airport near train station or get down at train station to buy bus tickets. We recommend taking a 48 hr or 72 hour pass so that it is easy and you dont have to worry much.
You also have direct train option from the Fiumicino airport to major train junctions, but it is about 12 Euro, so kind of expensive I think, so we took the bus.
One of the things you can do is print out Rome Bus Map beforehand and look at the maps for buses that can go to your place. Also, you can use Google Offline Maps to have your last directions to have the transit info saved.