Best Way to Get from Rome Airport to City (Pantheon Area)?

Taxi at the taxi stand.
48€ flat fare all inclusive from airport to anywhere in the city center.
The train station is not walking distance from the pantheon, you would need a taxi from the station if you have luggage, so in the end train+taxi wouldn't be any cheaper if there are two (or more) people. The train alone is 15€ pp.

Simple way would be to take the train -Leonardo Express - from the airport to Termini station. From there you could get a cab or, if you are feeling adventurous, the 64 bus to Argentina.

Do you want cheap? Easy? Fast? How many people?

The standard route would be the Leonardo Express into Termini, then either Taxi or Bus to the general area. (30 to 40 euro for two)

You could also take a cheaper train to the Trastevere Station, Tram to Largo Argentina, then walk or taxi to your hotel (~20 euro maybe for two)

Or you can take a taxi or car service direct to your hotel from the airport for 48 euro for up to 4 people

For Uber, I would verify service. Rome tried to ban it earlier this year, not sure if it stuck.

Since you asked for the best way (and it's not always the cheapest), consider a private car service. After a long flight, some jet lag, not being familiar with the area, there is nothing better than having your private driver meet you inside the terminal and whisk you to your hotel. Especially since you'll have your 70 year old mom with you. No fending off the "independent" cabs, waiting in the regular taxi line, or just having a "lost" feeling. It is not the cheapest way, but it is definitely the best. We can not say enough good things about Rome Cabs. It is a private car service (not a cab company) but only cost about 10 Euro more than a regular taxi. We always use them and have always been very pleased. Go to romecabs.com and see what you think. We usually send an email and they answer us with a quote within 24 hours. You do have to pay cash (Euros) and the end of the journey but not in advance.

I used Roma Shuttle http://www.romashuttle.com/ They meet you at the designated location which was easy to find, take you to the car and already know where you are going because you told them in advance (online form). 45 euros each way for 1-2 people, add 5 euros for each additional. So it depends on how many people, what you are willing to spend and what works best for you. I know many people say take the train and all that but let's face it, after an overnight flight the idea of me schlepping my stuff on a train and sorting all that out may save me a few bucks but I'm on vacation, and I may be a cheapskate most of the time, but 45 euros is ok with me. Sometimes convenience is just worth it to me. Already scheduled romashuttle again this year.

Ask your landlord for a recommendation? Last time we were in Rome our hotel arranged for a driver to pick us up. Hotel had a 'special' for that service, provided one stayed a minimum number of days.

You may not get a free ride , but the landlord could have a suggestion that makes for a smooth, no fuss transit.

For that location with luggage a fixed rate taxi I think makes the most sense.
I have done that previously ; for an upcoming trip the hotel offered to handle the arrangements for us for a private car service and the total R/T amount was 110 euro so only a few euro more each way than a taxi which I thought worth it so had them sign me up for that.

I am sure they are just contracting out a 3rd party car service but it was easy and costs me less than doing that on my own to let them make the booking.

We are using http://bobslimos.com/. Only €65 for 6 adults and our luggage. Our hotel is located near the main Termini. Not the cheapest, but divided by 6, not too expensive. And after a 9 hour flight, more relaxing.....

I second the suggestion to use Romecabs. I have used them several times and they are fantastic. They are completely reliable and professional. Like the other poster indicated, they are not much more than a regular cab but are way better. They use clean and comfortable cars with A/C whereas several cabs I took in Rome did not have A/C.

We always go to the taxi stand at the airport upon arrival and pay the fixed rate which now appears to be 48 Euros. If we are flying out of Rome and have stayed at a hotel we just have them arrange transportation to the airport. If we have rented an apartment, I have been able to arrange transportation to the airport when securing the rental.