Train all the way. Rome to Venice takes 4.5hrs. Venice to Vernazza takes 7.5hrs. Vernazza to Siena takes 4.5hrs. Siena to Sorrento will take 7.5hrs (including the private Circumvesuviana train from Naples to Sorrento) and finally, Sorrento to Rome takes 3.0hrs. There are no airports close to Vernazza, Siena or Sorrento.
One alternative for the Vernazza to Siena segment is to take the train into Florence and then take the SITA Rapide (Express) bus to Siena. The bus takes you to the city wall. The Siena train station is about 1km uphill from the train station but there are frequent local buses to the city wall. Your choice.