There is an average of three trains/hour from Civitavecchia to Rome Termini. Regional trains take about 1hr15min and the fare is 4.50€. ICplus trains take about 45min and the fare is 8.50€.
To quote a note on another thread from Larry of Elk Grove, CA: "Cruise and ferry passengers can catch a shuttle to the entrance of the port and walk four blocks to the train station. There are taxis there but they are overwhelmed by the departing passengers. Also, they don't want to take you to the train station, no money there. They are looking for the trip into Rome."
Villefranche-sur-Mer to Nice or Monaco: frequent and cheap bus and train service. Villefranche to Eze-le-Village: bus service is less frequent but adequate. Here is a link to a page showing the timetables and map for bus route 112 that goes from Nice to Beausoleil with stops in Villefranche and Eze-le-Village.