The major islands in Italy - the regions of Sicily and Sardinia - both have relatively limited train service once you're on the islands, but only Sicily has train service from mainland Italy.
The ferry crossing between mainland Italy and Sicily is so short that trains roll aboard the ferries on one side and roll off on the other side. To reach Sardinia, you must get on a ferry either by car or walking on.
The main island of Venice is connected to the Italian train network, but the other islands of the lagoon are not. Most other islands throughout Italy are not connected by train to the mainland, nor do they have train service once you're on the island.