From Rome airport (FCO) to a train station near Colosseum

You can take the Leonardo Express from FCO to Termini Train Station, then take a taxi from Termini to whatever destination you want.

Is it better to get a taxi?

Probably.

Are you alone or have others traveling with you? If you have others with you the cost per person for a taxi gets better.

Taxi from FCO to city center is fixed rate 55 euros

Leonardo Express is 14 or 15 euros? then a taxi from Termini to near Colosseum probably no more than 10 euros

Be sure you get in the official taxi rank in front of the station or in front of the airport

You can also take the Rome Metro from Termini to Colosso station, just two stops. The Colosso station is on the north side of the Colosseum and just across the street. IDK how far that puts you from your hotel, also how much baggage you will have. The trains are frequent and as I recall Termini is easy enough to navigate.

With luggage and not being familiar with Termini and the Metro, I would suggest either taking Leonardo Express to Termini, then a taxi to your hotel. Or if there are two or more, just take a taxi from the airport. You will be frazzled if you are coming from a long flight and I would not tackle the metro under those conditions.

If you are coming off an international flight and have luggage…. take a taxi, in my opinion.

I normally take the Leonardo Express from FCO to Roma Termini, and then taxi to my hotel. There's a taxi rank right in front of the station so you won't have any trouble finding one.

As you haven't taken a train in Italy before, it would be a good idea to do some research on the subject prior to your trip. Several trains stop at FCO so be sure to board the Leonardo Express. You may find this video helpful - https://www.youtube.com/watch?v=41fTAUbsAuo .

If you'll be using the high speed trains between cities, note that tickets are specific to train, date an departure time so can only be used on the train listed on the ticket. Tickets for the fast trains include compulsory seat reservations. This video is a bit dated but still provides some good information - https://www.youtube.com/watch?v=Q7whfEiZp34 .

Note that at Termini there will be scammers trying to lure you into gypsy cabs which will cost many times the normal rate. coming in exhausted one time I walked out front and someone said 'taxi' and I followed them to the curb, then asked the driver what the cost would be to my address and they said 40 Euro -- I knew it should be closer to 10 so I backed up and looked around and found the official queue which then was a ways off to the right. If it is now right in front of the station that is better -- but be sure you get in line in the official queue and do not allow yourself to be put in a 'taxi' by someone approaching you with 'need a taxi?' -- these are always crooks.

Thank you all for your replies. I should have stated my plans in more detail. My friend and I (2 senior females) arrive in FCO at 10:30 AM, early December weekday, then check in (I mean leave our luggage) at Hilton Rome Airport which is connected to the airport, then after some time head out to visit the Colosseum. We will be coming back to the hotel afterward. We meet the rest of our tour group next morning at the airport to fly to southern Italy.
So, to get to the Colosseum, it looks like I would either do Leonardo Express train + taxi (since I don't probably want to deal with their metro) or taxi alone. If we do taxi alone, it is fixed 55 euros to city center, so it would probably be more to Colosseum?

Now, to return to my airport hotel, do you know if there is a taxi stand by the Colosseum?

I read that Uber in Italy is not like the Uber in the U.S. - this internet article I read was ~2 yrs old, so we better not even think of using an Uber. Do you know this is true?

I read that Uber in Italy is not like the Uber in the U.S.

That is correct.

The Colosseum is within the flat rate taxi zone ( Aurelian Walls)