My 14 day itinerary

Looks very good. Good balance. Mount Vesuvius (personal opinion) is just smoking hole in the ground but it is important for some people. I would spend a half day at the museum in Naples where most of the good stuff from Pompeii is exhibited.

There is no ferry in Praiano
Your transportation will be by bus which can get very very crowded, also very nausea inducing even if you are not prone to motion sickness

I’d pick a town with a ferry
We preferred Amalfi

Another suggestion is to put all Rome nights at end, travel right to Amalfi area on arrival
Eliminates that split stay/ 1 nighter

You know, when I look at that itinerary, it seems awfully concentrated in coastal/island locations with not a lot of things to do except look at and walk through pretty scenery and cute villages. I like doing those things myself, and I don't doubt that every day, taken alone, would be very pleasant, but I'd find 8 days out of 9 (discounting the Pompeii/Vesuvius day) doing those things to be too same-y on a short trip. This may be just the right itinerary for you, though.

Full disclosure: I've been planning a trip of my own that may well include nine days in that same area. The difference is that it will be part of a longer trip with a lot of city time, including probably at least a full week in Naples and more than a week in Rome--plus time elsewhere. I'm also considering making one of my bases Salerno rather than Sorrento.

If your time permits and if you are interested in more ruins, a half day in Herculaneum is nice. It’s a very different site from Pompeii and good to compare the two. Also a visit to the Archaeological Museum in Naples to see all the goodies that were removed from Pompeii; most of what you see at the site are reproductions.

Totally agree with Christine, go to the Amalfi coast first. We too prefer Amalfi, it’s the hub of the coast for transportation. If you want a smaller town, check out Atrani. Cute fishing village, has a small beach, great restaurants and at most a 15 minute walk to Amalfi town.

Next I would do Ischia by ferry from Amalfi. While the Porto area is very busy, look at the other villages. One of our favorites is Forio. Very nice beaches. Bus service from here to other towns is quick and easy. Do recommend the Poseidon spa. And the natural waters at Sorgeto beach.

https://www.ischiareview.com/forio.html

From Ischia take the ferry to Capri, then to Sorrento.

Leaving all your nights in Rome, eliminating a hotel change.

Edit. This is Praiano!!

https://iowagirleats.com/life-on-the-mediterranean-coast/

Yes, Rome once at the end; Amalfi first with driver, picked up at Napoli Afragola, train from Fiumicino Aeroporto, a little over 2 hours.

In Amalfi, the Residence Hotel Amalfi is right across the road from the port and bus stops. Good dinners at La Galea, up the street past the Duomo.

Much easier & faster to reach Amalfi this way.

Train, Roma Termini to Salerno about 1:40, then ferry 1 hour or bus to Amalfi. Ferry dock about 2 blocks from train station.