Train from Fiumicino to Roma Termini - revisited

It is the simplest solution. Direct, non-stop. There is also a bus option to Termini around 8 to 10 euro. I think the local train goes to Termini but makes a couple of stops. Didn't think a change was involved.

Leonardo Express. It is only 14 euro and direct non-stop. After a long flight who wants to find a bus or take a regional train. Well worth it. Save your efforts for getting to your hotel from Termini.

Is the Leonardo Express somewhat easy to find once I get through passport control?

The Leonardo Express train is very well marked with signs everywhere. Just be sure to validate your ticket in the yellow machine before getting on the train because your ticket will be checked in route. Fines are collected in Euros on the spot, and they're not cheap. It is fast and efficient.
I also suggest you research your accommodations and figure out how to get there from Termini. When in Rome, we try to use buses and the underground for most part--staying away from expensive taxi's. Last two trips, we stayed in the same familiar neighborhood.

For Termini, only the Leonardo Express goes to Termini, the other primary train, the local FL1 does not go to Termini. It does stop at Trastevere Station, Ostiense Station, and Tiburtina Station among others, so can be handy if you are staying in the Trastevere area for example. If you changed at one of those stations you could save a few bucks, but spend more time getting to Termini.

There are direct buses also for about 6 euro.

Depending how many in your party, a taxi may be a reasonable cost effective alternative. It is a fixed rate to the typical tourist destinations "within the walls" of central Rome

The ticket hall can be confusing. Private vendors say they sell train tickets but may try to persuade you to take a van instead. Allow enough time to search out the authorized ticket seller.