In addition to seeing the Amalfi Coast probably using local buses that you can get off, explore, then catch the next bus, we want to ss the Reggia di Caserta palace and gardens, Pompeii, Sorrento, Pompeii Museum, possibly sites of Naples (any 'don't miss' suggestions?). Is Mount Sevuius worth seeing (why)? We're thinking Sorrento would be a good base for a hotel. If only using public transportation, can we easily see all of this? Do we need local guides for any of it (which)? Any recommendations on hotels and local guides to use? Opinions of the Isle of Capri? YouTube videos did not look impressive for Capri.
I'd say from Sorrento you can quite easily go to Pompei and Naples by train, and to other places on the coast by bus. I think from there it will be harder to go to Caserta and Vesuvius. If you have time, you should also see the old town of Ercolano. I also like Procida, the only island of Naples Gulf that still remains romantic and (almost) like an old fisherman place. Of course Positano, as Ann says. And if you like walking, the "Path of Gods" from Agerola to Positano, with a wonderful sight on the sea, the coast and Capri, starting from 600 m. above the sea level.
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If you are based in Sorrento, I would recommend a side trip to Villa Cimbrone in Ravello, where the gardens are open to the public for a small fee. A most spectacular view can be had from the promenade, just breathtaking!. The Humphrey Bogart-Gina Lollobrigida film "Beat the Devil" used the gardens and promenade for some of its scenes. Gore Vidal, God rest his soul, lived nearby for many years and once wrote: "Twenty five years ago I was asked by an American magazine what was the most beautiful place that I had ever seen in all my travels and I said the view from the belvedere of the Villa Cimbrone on a bright winter's day when the sky and the sea were each so vividly blue that it was not possible to tell one from the other."
If you are adventurous, you might also consider Punto del Capo, where you can swim amidst Roman ruins known as the Baths of Regina Giovanna and then through a natural arch into the Gulf of Naples. It can be reached from Sorrento by following the main street Corso Italia and then up the hill along Via del Capo. It's a bit of a walk but well worth it.
When last in Sorrento, I stayed at the Hotel La Tonnerella and was very pleased with both the price and accommodations. Check it out at http://www.latonnarella.com.
We just got back from our vacation and we stayed in Amalfi we stayed at http://www.hotelexcelsioramalfi.com it had an amazing view, it was up higher on a hill and they have a shuttle bus that runs all day and until midnight ... its about a 5minute shuttle ride to the center of the town, they also have a private beach that the shuttle takes you to. They have a restaurant at the hotel and the people who work at the restaurant are so nice and pleasant.
If you are renting a car, I highly recommend you rent something like a Fiat, because the roads to get to any city on the Amalfi coast are enough to give you a crash course in being a good driver. We took a tour to Pompeii and Mount Vesuvious.If you are interested in hiking and seeing a volcano, than I recommend Mt Vesuvius. Make sure you wear actual hiking **SPAM**, the ground is kinda loose and it is easy to fall, especially on your way down. For me seeing Vesuvius was worth it, because I have never been at the top of a Volcano in my life, and I probably never will again, and it is also one of the most historically known. This is the company we used. you will see that they pickup in most of the major Amalfi coast cities : http://www.sunland.it/en/excursions-8/d ... ius-3.aspx
Something that could save you money is the Campania arte card, it is a card that you can use for buses, museums , and other attractions in the Campania region of Italy: http://www.campaniartecard.it/EN/index.cfm