If you are flying into Rome airport there will be a reprsentative from your cruise line waiting for you and other passengers they will escort you to the coach which will take you to the port Regards Irn.
We did this journey flying with Easyjet and therefore arrived at Ciampino airport.
We took the the local bus to the train station (a 20 minute wait - ten minute ride), then take the train into Termini (half hour ride to central Rome) and another train (one hour ride) to the port.
The ship will be about ten minutes walk from the station at Civitavecchia (turn right as you leave the station, keeping the sea on your left. Walk straight ahead to the port).
Your carrier may well offer transfers into central Rome.
You may be flying with another carrier and arrive at Fiumicino airport and I regret I cannot offer advice regarding arrival here, however the site for railway information is 'trenitalia.it' which should provide information on routes and timetables.
The ship will offer transfers but it will obviously less to do the journey yourselves.
Finally, you could think about hiring a car. It is around an hour/90 minutes to drive from the airport to Civitavecchia where there are several hire car companie,s so you could arrange to drop the car off there.
We did the Rome transfer to Legend this year. We flew into Ciampino airport and caught a shuttle bus with terravision to Rome. Tickets can be bought either on the plane or on the bus - 8 euros per person. You will find the buses just outside of the arrivals door. The bus takes you to Rome main train station. From here buy a ticket to the port station I can't remember the exact price, but I think it was about 6 euros per person, (they run approx every hour). You can buy a ticket from a ticket desk but there was a very long wait, so we got ours from the ticket machines. They have a English option and it was very easy to get the ticket this way. The train was very good quality. On arriving at the other end approx 1 hour later, we then walked to the ship. This was a mistake as the Legend was docked at the far end of the port and it took us over an hour to walk with our cases in the hot sun. We later found out we should have walked from the station ( a 10 min walk)to just inside the dock gates and on our left we would have found a free shuttle bus service that would have taken us directly to the ship. Or yu can get a taxi from the station to the ship there were quite a lot of taxis waiting outside the station, but I have no idea on the price of this option. I hope this helps. The Legend was a fantastic holiday for us this year the best we have ever had. We actually flew to Rome a day early and stayed in a B&B near the main railway station and spent the day exploring Rome. This was well worth doing and a lot cheaper than going on a Royal Caribbean organised tour.
I assume that you are doing flights independently - you can buy the RCI transfers for £57pp each way......alternatively you can catch the train into Rome for €13 and then the train down to Civitavecchia for €9 which is easy [and cheaper] to d.
Keith Boyd's answer is accurate, if you have booked flights and transfers with your cruise.
Many people choose to make independent travel arrangements (perhaps arriving a few days early and incorporating a stay in Rome) and therefore have to make their own way to the ship.
Providing one leaves plenty of time for this transfer, it is a fun thing to do! Italian trains are air conditioned and we think they are wonderful (run to timetable and cheap and easy to use etc.).
I do understand that where passengers are perhaps a little frail or elderly, they would be better served by reserving flights and transfers from the cruise company.
Or....as August is hot and you will have heavy suitcases, go on the net and have a look at Rome taxis...you can get an air-conditioned chauffer-driven Mercedes for €120, per limo not per person (about £80 ) and arrive in style, comfort and relaxed...go on spoil yourselves you'll be on holiday and you did say THE BEST WAY....Ida Dow.