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My husband and I caught a train into central Rome from just outside the cruise terminal. It cost us (last year) about 10 euro each way which was MUCH cheaper that the ship's tour. There are plenty of trains although they can be a bit crowded later in the day but we had a lovely long day in Rome.
Most people take the train but if you opt by the highway:
Civitavecchia is an hour or more by car, bus or train from Rome and I would not recommend doing a tour on your own from that port city. You can take a train, van service, or a taxi [100 Eurs] but its a good hours drive on a highway and you can get caught in traffic. That could delay you getting to the city and also getting back to the ship. On our tour we were an hour late getting back to the ship and missed dinner. But since we were on a ship sponsored tour the ship waited for us. If you go on your own the ship will not wait for you. So by far the train is the safest, or go with a ship's tour which normally allows you plenty of free time to wander about at will.
Take the train it is the easiest and quickest and the most cheapest - don't understand how these cruise lines can charge their customers a fortune for a trip that you can do for a few Euros.
We found the shopping people to be helpful in providing the train times and maps and then made our won way.
We were also told not to take the last train as they can get cancelled and the ship will not wait for you.
Take the shuttle bus to the port exit, it can be a long walk. Station is about 5 minutes walk along the sea front. Return tickets to Rome are about 10 euros (it seems to vary according to who serves you), you must time stamp the tickets using the yellow box on the platform, you can be fined on the spot if you don't and you are caught. The tickets are also valid for the subway in Rome when you get there. If you get off at San Pietro station (about 1 hour 10-15 mins) you can see the dome of St Peter's and just follow the crowds to St Peter's square. There is no entry fee to the basilica but you can pay to go up to the dome which has an incredible view over Rome, go to the cupola at the very top. For the Sistine chapel you have to go round the back of St. Peters (to the right); it's not immediately obvious. When you have finished there it is possible to see the rest of the main sights, at least externally, in a day but a fair bit of walking is involved. Taxis can be useful if you get stuck "out of position". Free maps are readily available, try in a McDonalds. You can get the train back from Rome Terminii (about 1 hour 20-25 min) but you have to get the subway to it from the Colosseum area, the trains from Terminii leave from a distant platform usually; it's a good walk so leave time. You can get train timetables from the Trenitalia website (in English), just google it; it's also worth looking at Google Earth to get an aerial view of the city and work out where you want to go and in what order, the Trevi fountain for instance is tucked away in a back street. As others have said make sure you leave enough time to get back, at least one train before the last one you can get. The other main drawback is the heat, it can be seriously hot in July/August and if you are trekking about it can be very tiring, make sure you stay hydrated. It is well worth doing yourself rather than taking a tour, you can book tickets online for the Colosseum now I understand. Don't hand your camera over for the fancy dressed centurions at the Colosseum to take your picture with one of them unless you fancy paying a fortune to get it back.......
Take the 'free' shuttle bus that will be waiting along side the ship - the cruiseline will try and sell you shuttle transfers and will not tell you there is a free service laid on by the port!
Many thanks everyone for your advice.