FCO to Trastevere

I would take a taxi from the airport.
48€ flat fare, and you are there in 25-30 min.

It's pretty expensive.

The FL1 airport train costs 8 € per person and calls at Roma Trastevere station, but I have no idea about the distance to Villa Farnesina and how to get there from Trastevere station.

Train tickets must be validated before boarding the train.

Take the FL1 train from the airport to Trastevere, then the #8 tram from the trastevere train station to the last stop before it crosses the river, Belli I think. Don't freak if you miss the stop, just get off at the next one walk back across the bridge. Your destination is about a ten minute walk on level ground. Consult google maps, it helps. You can buy the tram ticket in the train station at the tabacchi shop.

After 1900 the fare goes to €60 for the taxi.

As a couple of people have mentioned, the FL1 train is what you need. Ticket is cheap, make sure you validate before you get on. There are a number of stops, takes about a half hour to get to Trastevere Station. The last train leaves FCO about 10:30 PM, so you should be good.

Once at the station, head out the front entrance, through the parking lot, and off to the left on the main street (Via Trastevere) and you will see the tram stop. I can't recall if the ticket from the airport is also good for the tram (a Metro ticket is good on the train, but not to the airport) but you can pick up a metro ticket in the station (1.50 euro). As already mentioned, get off before the river and walk up the river to your hotel.

We took the train from the airport to Roma Trastevere station and then a taxi.

Where does it say that after 1900 (7pm) the taxi fare goes to 60€?
No posting fake news please.

48€ may not be cheap but I wouldn't skimp on the convenience after a long flight especially if you have luggage. The train takes you to the Trastevere station for 16€ but the Villa Farnesina is not walking distance. You can take the 8 tramway but it's not that frequent in the evening and once you get off you still have a 1 km walk (15 min). I hope you travel very light. That's all I have to say.

Well I've watched some YouTube videos and it looks pretty easy to navigate, and having some feedback definitely gives me more confidence. Also is it free to travel with a child who is 9 on bus or train? I can't find that anywhere on the sites or forums. I know I'm London it is:) As for the long trip, we're only flying in from Paris and we arrive at 8pm. If we aren't on time for whatever reason, benit delays or child needs to go to bathroom or whatever interruption on our path, we'll resort to a taxi. If I can spend less than €25 for two travellers I am happy! The train is €18? Is that what I read? I can't see the previous comments on my phone:( All of this is helpful for planning. I've been planning for months and months and I'm only 14 days from leaving! So thank you! We are not at the location that's just near it, we're fine walking 20 minutes from a bus. We live by bus here in Victoria, BC. It's how we go. We rarely take taxis here unless we're out past the last bus. Hardly happens.

If you'll just give us the actual name & address of the place, we'll see what kind of walk. Be aware that it could be up a steep hill in the area of the Villa Farnesina. And the train is 8 EUR per person.