9-days Rome and Amalfi Coast-First timers

Day 10 Depart Naples 10:am

Be sure to be in Naples the night before

I would be inclined to give Rome more time than Sorrento/Amalfi Coast area.

Paestum is best/ easiest visited from Salerno
You could train from Rome directly to Salerno, stay maybe 2 nights and visit Paestum ( trying to get to Paestum that day might be a crunch)
Then ferry, bus or driver to Sorrento

https://community.ricksteves.com/travel-forum/italy/amalfi-coast-for-7-nights-itinerary-help

https://community.ricksteves.com/travel-forum/italy/herculaneum-and-pompeii-questions

https://community.ricksteves.com/travel-forum/italy/amalfi-coast-for-5-days-where-to-set-base-from

https://community.ricksteves.com/travel-forum/italy/how-do-i-get-from-pompeii-to-positano-is-there-a-direct-route

https://community.ricksteves.com/travel-forum/trip-reports/surviving-naples

https://community.ricksteves.com/travel-forum/italy/amalfi-coast-day-trip

We disliked Naples. Positano was our last leg, and we decided to go to Naples the last day and night before departure. We so wish we had just taken the fast train back to Rome the last day.

I think your plan is fine.. 5 nights Rome, 3 nights Sorrento and 1 night Naples before departing.

I actually think your plan is perfect. Rome is hectic and frustrating, although incredible. Three full days + is perfect! (Vatican, Coliseum and Forum, Pantheon and Spanish Steps, etc.) We really enjoyed our time in Sorrento. Four full days would allow you to see: Capri, Pompei and Vesusuvious, Almafi Coast, and Naples. Rick says Italy is more charming outside of Rome, and we 100% agree!

I second AmandaR's assessment, great 1st time plan to Italy, wish it was me. Tip, think nights instead of days for accounting for your time abroad. If you enjoy and appreciate big cities, Rome is truly a wonder. The history, the monuments, the sites, the citizens -- there's nowhere else like it. I think 5 nights meaning 4 actual days is an excellent if short visit to the Eternal City. Definitely hook yourself up with a local guide for the superior insights and details expert commentary provides on what you've traveled so far to see and experience. Being on the coast Sorrento is supreme in location and who doesn't love a lemon grove in their downtown? I personally would stay 2 nights in Sorrento and then finish with an intense southern Italian flourish with 2 nights in Naples. Naples astonishes in how successfully people live so compactly with, on and around one another. You absolutely need to keep elbows in as a pedestrian which can be quite the shock to the system. The region's incredible cuisine (bonus points for seafood lovers) is blessed and rife with ultra fresh ingredients thanks to the volcanic soil producing fabulous fruits, veggies and floral. You must go to Museo di Capodimonte and you should go to Castel Sant'Elmo for the incredible view of the former kingdom of Naples. I love Italy and I love southern Italy more, buon viaggio.

It depends how much you like cities. We loved Rome and I would give it the 5 nights. We spent 4 nights on the Amalfi Coast and 8 nights in Rome and that was a perfect balance for us. We never ran out of things to do. If you think the intensity of Rome might wear on you, then give the extra night to Sorrento.

You need a minimum of four nights in Rome and add another night if you day trip to Orvieto (1h 15m from the Roma Termini station). I would then take a direct train as Christine suggested from Roma Termini to Salerno (1h 45m) and sleep there two nights to visit Paestum.
You can then take ferries from Salerno to Positano and sleep there one night before taking another ferry from Positano to Sorrento for three nights that makes a good base to visit Capri and Pompeii. From Sorrento take a ferry to Naples.