A blank canvas? Rome south? Holy opportunity, Batman, let me count the ways, but here's a somewhat different one, involving the eastern Italian coast and Sicily, given you have a month (destination/# nights in bold):
Arrive Rome
Rome 3
(train Rome/Salerno)
Salerno 3 [visit Pompei, Naples]
(train Salerno/Matera)
Matera 1
(train Matera/Bari)
Bari 3 [visit Polignano a Mare]
(fly Bari/Palermo)
Palermo 5
(bus/train Palermo/Taormina)
Taormina 3
(train Taormina/Messina)
(train on ferry Messina/Reggio Calabria)
Reggio Calabria 1
(train R.Calabria/Paestum)
Paestum 1
(train Paestum/Amalfi)
Amalfi 5 [visit Positano, Sorrento, Ischia]
(bus/train Amalfi/Rome)
Rome 5
Depart Rome
Now that I look at it, this looks a tad insane, but I'd take this trip. I do love Italian train travel but the downside in southern Italy is that you are beholden to their schedules, which are not always 100% dependable like the north. I've always wanted to do the train on the ferry from Sicily to the mainland, as well as visit the caves of Matera and the east coast with Bari and Polignano a Mare, which is gorgeous.
And 3 days in Rome at the outset of your trip to get acclimated, then 5 days in Rome before you fly home is a luxury that should be savored. It's such a great place to chill & reflect, even with so much around.
FYI none of the train travel here is more than 4 hours in a day, day trips all less than an hour from a base, but there are a couple of 1 night stays as you see. And for a month in Italy--you have to see Sicily, which requires one short flight. It's that good.
Take this all with fun and a grain of salt, and enjoy your planning!