I have done 2 cruises one in the med and one in the caribbean for those we did a mixture of our own excursions and those arranged by the cruise line, we researched on the web before hand each port of call to see if there was anything close by we wanted to do and then we got a taxi (or sometimes there was a free shuttle service the cruise line provided to reach the centre of the ports town which was free) when we got a taxi (they are generally plenty around we negotiated a price before we took the taxi so we got to see the sights at a reasonable price. Then we compared the excursions the cruise line offered and took those that we couldn't miss!
My other piece of advice seeing that you are going to Rome is that port is about 1 1/2 hours from the centre so excursion royal caribbean provide would be better.
The excursion to rome from Civitavecchia is not worth paying for. Civitavecchia has a train station and there are trains to Rome at least once an hour. the return fare varies, but starts at about 9 euros, depending on which train you take.
Try typing something like Italian train service into a search engine and you will soon come up with the train info.