Lucca to Solerno

Train from Lucca to Florence, then another train from Florence to Salerno (sometimes another change in Naples required).
If you can depart from Lucca with the 8:31 train to Florence, you can make it to Salerno in 5h 11min with one change only (in Florence).
Use www.trenitalia.com for full schedule.
Use a date in the next 7 days as the regionale timetable comes out only 1 week prior (from Lucca to Florence you will be on a regionale train).
Flying is not an option, There are no flights to Naples from Pisa (the airport closest to Lucca). Besides, even if there were a flight, by the time you go to the Pisa airport, check in, go through security, fly, then transfer from NAP airport to the station, take the train from Naples to Salerno, it will be way longer than the train option directly from Lucca to Salerno.
Driving is another option, if you happen to have a car while in Lucca. However driving won't be any faster than the train, au contraire. It's all freeway, but it's almost 600km (370 miles), therefore well over 5 hours' drive plus pit stops.

If your destination is Salerno, then I'd follow Roberto's advice for the train.

Thank you very much! One more question please! When we leave Salerno for Rome would train also be the best option and from where Salerno or Naples?

"When we leave Salerno for Rome would train also be the best option"

Yes, train is best. Use the Trenitalia website to find schedules, following Roberto's advice about looking in the next 7 days to see all trains.

"and from where Salerno or Naples?"

Not sure I understand the question. Trains go from Salerno through Naples to Rome (sometimes direct, sometimes with a change in Naples). So, if you are in Salerno, take the train from there to Rome. If you're in Naples, take it from there to Rome.