I'll let others address the Pompeii trip since I visited Pompeii from Naples.
For the Amalfi Coast, taking a Freccia train to Salerno and then a ferry along the Amalfi Coast will probably mean the least travel time (if you count the ferry time as sightseeing rather than pure transportation). A bus from Rome would take way too long. I haven't done this, so I don't have a good sense of how much time will be lost to connections between train and ferry or waiting for the next ferry when you hop off to see a town on foot. We probably need more specific info on the timing of this trip; I think ferry schedules from Salerno could be very different in late March vs. late June.
Trenitalia.com will give you rail schedules between Roma ( Tutte le Stazioni ) and Salerno. The fastest trains take 95 to 126 minutes each way, but those aren't super-frequent.
Travelmar.it has ferry schedules.
I'm not sure you'll end up with very much feet-on-the-ground time on the Amalfi Coast. I hope your trip is in May or June when days are longer.
I know you really want to do this, but that doesn't change the geography. Four nights in Rome is only 3 full days. You may have a few additional hours on the day you arrive, but if Rome is your landing point after a transatlantic flight, one or all of you may be zombies that first (partial) day. You're proposing to give up one of your two full days in Rome in exchange for a few hours seeing the Amalfi Coast. Perhaps you've been to Rome before?? If not, I can't imagine you'll be very satisfied with what you can accomplish in Rome in one day.