Steff, you need to give the month of the year to get the best answers. All of the good advice you have already received is affected by the massive crowding of this area from May to September. Ferry and bus schedules have high and low seasons, so what you can do and how long it takes to wait for public transportation are also affected. All the destinations (except, maybe Salerno, where I've never been) book up at least six months in advance. In Sorrento or Positano, you can expect to get, at best, your fifth choice of housing.
I'm not sorry we spent five nights in Sorrento. Although I found it a purpose-built, postwar, reinforced concrete resort city (with a magnificent view from the high-priced hotels on the cliffside (we stayed in one, the Ambasciatori, which Rick calls "a worthwhile splurge" - but it's even more expensive than that.)) Having been so negative, I would have to say that Sorrento suffers less from the worldwide trend so visible in Positano and Capri, of the "usual suspects" of multinational luxury goods vendors. I mean, just how many Cartier boutiques do I need to see in my liftime? Even Angkor, Cambodia (because of Chinese tourism) now has them!
I don't like changing hotels too often. But I'd observe that Naples has so much to see (including a daytrip to Caserta palace if you have the time) that you might sleep there a night or two. Even Sorrento is a long ride on the train. BTW, while I am willing to consider that Salerno may be an attractive modern place to stay, I don't accept the encomium of "fast trains." I think what that post means is that the Frecciarossa service sometimes might let you ride direct from Rome or even farther away, to Salerno. A "one-seat ride" is, in fact, worth something. But that does not mean that Salerno is the best place to see Pompeii and Capri from. ("best" is an opinion, of course.)
Having brought up the issue of public transportation, your travel budget is very important. If you need to use the cheapest means to get to the actual AC (your first three cities listed), you are going to spend a average of 45 minutes in line in Sorrento, outside the Circumvesuviana station, waiting for the bus. It's hot in the sunshine. You may have to stand on the bus, too, and only one side has the famous view. Our host, Rick, has noted that those visiting Positano and Amalfi from Sorrento on the same day should see Positano first, because the return bus originates in Amalfi, and in the afternoon, it may be so full that it doesn't even try to stop in Positano. Get my drift?
We hired a private car and driver to see your first three towns, but that runs 240 Euros or more, which not everyone can afford. BTW, we decided to skip the Blue Grotto on our Capri day, because we've seen another grotto in Croatia. This can be a huge time waster, and there is much else to do on Capri. Note that Capri is HIGHLY pedestrianized, and you should expect to walk 2 to 4 miles in a "successful" Capri day. I found the microscopic public busses on Capri to be worthless and to have crushing waiting lines. We took a cab as far as the cab could go, to save time. We walked back (i.e. DOWN) to the ferry on the Phoenecian Steps, but that's too hot and rugged for everyone.