For €40 you and your granddaughter can be met at the baggage area of FCO and taken directly to your hotel in a clean sedan with an English speaking driver.
This is the car service I've already informed you about and that is highly recommended by many. You don't have to prepay or provide a credit card to make the reservation and you can pay the driver in cash or with a credit card:
http://www.romeshuttlelimousine.com/...t-transfer.php
I do suggest you take this easy way - and for 2 people the cost is most likely equivalent to taking the express train plus a cab from Termini.
I understand wanting to stick to a budget, I do the same thing. But this is one area I don't scrimp. Arriving into FCO from the U.S. I'm exhausted (and cranky usually!) and pushing through the hordes of people to get to the train station, taking the train -which can be packed - and having to remain vigilant on the security of my belongings, and then surfacing at Termini - a truly crowded, crazy making place, then trying to find the official cabs vs the gypsy cabs, and lastly hoping the driver doesn't take me on the scenic route with his meter set to the wrong zone is just not worth saving a couple of bucks. IMO.
But if you're not budgeted for €40 for transportation into the city then do a search here for the threads that detail getting from FCO to the city. (You also have the choice of taking the FR1 train for only €8 or the SITBus shuttle, likewise €8, or you even could take a Cotral Bus for even less I think).
RinR's site also has tons of info regarding transportion options. If you don't know how to do a search here, let us know, I'll be happy to give you instructions.
So what hotel did you choose?