There is a regular bus service from Fiuggi to the Anagni train station.
from there trains leave every hour to the main train station in Rome city centre (Termini Station) the journey takes one hour.
The cost of the ticket is 9 euro which includes bus/train return, and use of all of the public transport including the underground in Rome all day.
There are buses in the morning and late afternoon. from Fiuggi to Rome and back by motorway. The journey takes 90minutes. Less if the traffic on the ring road isn’t too bad.
There is also a bus service from Fiuggi to Rome that visits lots of little towns on the way, interesting if you want to see places off the beaten track.
If you have car hire there are trains which dont have a bus connection to Fiuggi. you could leave the car at the local train station in Anagni and take the train into Rome. the first train in the morning is at 4.55 the last train back 24.00.
If you stay in Fiuggi and you want to go Rome or vice versa, there are only a few opportunities of transportation.
One of them is going directly Termini by bus. Then you can choose metro or walking to reach your main destination. Ticket of this bus also can be used in metro. It takes approximately one hour (or 80 minutes) to reach Termini.
The other ways is going Anagnina by bus and taking the red metro line to city center. But I don’t recommend you this way. The mountain road has many bends and there are many bus stops in every village.
And the last choice is going to Anagni (not Anagnina) by bus and taking the speed train to Termini. I did not try this way.
There is a bar in the bus stop of Fiuggi. If you want to go somewhere it is the place for buying tickets. The employees don’t speak English but thanks to the timetables on the walls, you can understand each other by pointing that you want to say.
And be aware that there are so few busses ever day. The last bus from Rome to Fiuggi was about 7:00 pm. After then you have to find a taxi from Anagni, the last stop of speed train.