What is the cheapest way of getting to Rome city centre from Leonardo da Vinci airport and how long would it take?
I read about a train and an old line, but I am not sure how that works.
Cheapest there are the buses by Terravision and a couple of other bus companies. If you book in advance you can get a one-way for as cheap as 4€. For info and booking see the Terravision website. It takes 55 minutes. Note that here you are not flexible with your departure time. Buying tickets at the booth is possible but at higher prices and may sell out, especially in summer. In the latter case, don't expect any service, just take the train:
The most convenient train is the Leonardo Express train which takes you direct to Roma Termini station, departing every 30 minutes and taking 31 minutes. The cost is 14€ and a commission may be charged if you buy it at the booth. You have to convalidate it before entering the platform. This train is supposed to run even in case of railway strikes.
There is also cheaper regional trains to Roma Tiburtina station for 8€, every 30 minutes and taking 48 minutes. Both Termini and Tiburtina are connected to the metro network of Rome. You have to convalidate your ticket. For more info see the Trenitalia website.