I stayed at the Sorrento Hilton for a conference. If you have a high up room facing the town, the view is spectacular. If not, there are public areas where you can enjoy the view.

I took the train from Rome to Naples, then the Circumvesuviana train to Sorrento - I could not find anything more direct. The train to Naples was easy and got in early. But the Circumvesuviana is a (slowish, sometimes crowded) commuter train system separate from the intercity lines. Construction when I was there meant a long walk to the CV entrance, and no elevators or escalators. I had a lot of stuff for the conference so it was a pain. And be warned - in Naples guys will dress in light blue jackets and act like porters but there are no official ones - they will charge an outrageous amount to handle bags. Hopefully since this is a vacation you will pack light and have stuff you can easily handle. (I considered the ferry but was unsure about getting to it with all my bags.) But it very convenient for day trips to Pompeii or Naples from Sorrento and somewhat scenic.

When you get to Sorrento, you will probably want to take a taxi to the Hilton as it is a distance uphill from the train station (or the ferry dock). It was around 15 Euros for me and my luggage. Without luggage, it is a bit of a walk back to the center of town, but not too bad.

Four days should be enough for the places you want to visit. You can easily spend an entire day at Pompeii, but allow at least 2-3 hours for the highlights. I squeezed in a quick trip to Capri and due to a change in ferry times wound up taking the ferry back at sunset. It was beautiful.

Another tip: Look up the Roma Pass and the Campania Artecard before you leave (the pass rules may change between now and when you actually leave). They may or may not save you much money, depending on what you visit, but they can save time. For example, I bypassed a looong line at the Coliseum ticket window near closing time, giving me at least 15 more minutes to explore.