Is it worth buying the Trenitalia Pass to travel around southern Italy?

It depends on how much you plan to move around and for how many days. If you're in the south for a week and think you'll take the train 4-5 times, it might be worth it, especially if you include long routes like Naples-Palermo. The problem is that regional trains in the south are slow and often cost very little individually, so if you're only making a few short trips you might end up spending less buying tickets one by one. Check the pass prices carefully for the number of days you need and look at single fares before you decide.