Bus from FCO to Termini in Rome

Yes there are:

https://www.adr.it/bus-per-roma

Take the train

Yes, I took a bus last September.
I think it was about E9 or so, can’t remember.
There were a few different bus companies to choose from outside the airport and there was an electronic signboard with info about the stops each one made and what times they were all departing at.
I think I paid the driver?
It dropped passengers alongside of Termini on Via Giovanni Giolitti, though if all the work redoing the big open area outside the main station is done now, they may drop you there.

The regional train is 8,-, the Leonardo Express 14,-
Why would you want to take a bus?

The regional train is good for other stops, but does not go to Termini.

The bus is more economical than the express train at 6-8 euro. The time between the two is comparable, ~35 minutes for the train vs 45 minutes for the bus, however traffic for the bus is a risk. The express train may also be more flexible with frequent departures at least every half hour and as often as every 15 minutes. If there are more than 2 of you, or your lodging is not near Termini, a taxi is also an economical option.

The above in English:
https://www.adr.it/web/aeroporti-di-roma-en/pax-fco-bus

We have taken one of the coach buses from FCO to Termini and thought it was fine.

Where are you staying? The local train is good for the Trastevere neighborhood. We’ve also taken the tram 8 from there down towards Campo di Fiori and the Piazza Navona.