Without knowing specific dates and arrival/departure times for each city, it is really tough to comment further.
Catching a public transportation bus to catch a train is probably not going to happen.
Livorno to Genoa is 180km or about 2 hours drive. Livorno to Nice is 370km or about 4 hours drive. Rent-a-car and drive. Probably then only way to go.
Radiance of the Seas - Nov 2006
Legend of the Seas - Jun 2007
Costa Fortuna - Mar 2008
Legend of the Seas - Oct 2008
Celebrity Constellation - July 2009
MSC Orchestra - Jan 2009
Navigator of the Seas - March 2009
MSC Poesia - November 2009
Jewel of the Seas - June 2010