We just did 5 days on the Sorrentine Peninsula for our 30th anniversary and loved it. Train to Naples, a visit to the archaeology museum (where all the murals and mosaics and other art and objects found in Pompeii now are kept; they'll hold your bags too for free); then straight to the ferry to Sorrento for 2 nights. No regrets about skipping Naples, at least we saw it a bit from the taxi. Sorrento is the easiest base to get to Pompeii and has lovely vistas of the bay -- pick a place with a pool and view of Vesuvius. Not really beach-convenient because it's up on a cliff.
For Pompeii -- hire a personal guide just for your group. Antonio Somma whom Rick recommends was one of the best guides we had our whole trip. (pompeitour.com)
Plan at least one night in Capri, and ideally more. It is swarmed by day but after the last ferry leaves at 5:30-6 the character of the island relaxes and changes utterly. Hire a private boat with a captain to take you around on a tour; 3 hours goes by too fast, I wouldn't do less. The Blue Grotto is unreal but super touristy, and there are many other lovely spots and caves if you do a tour around the island. We rented a Vespa scooter overnight so we could ride to the far end and watch the sunset from the lighthouse. (On the coast, the sun sets early behind the mountains.) I would not advise that Capri be your first experience ever on a scooter -- the roads are steep and narrow with tight switchbacks up the mountain, though there are guardrails and no risk of flying off into oblivion. But it's not impossible -- it was only my second time and we managed fine.
The sole road on the whole Amalfi Coast is two lanes and perched on a cliff. Some of the most vertigo-inducing and beautiful views ever, especially if you sit on the right when it's eastbound. Buses are cheap and crowded -- with 10 people you may not have other good options other than a fleet of taxis. Beaches almost always require a climb down stairs and are pebble, so bring flip-flops and enjoy the sparkling clear azure water with no sand clouding visibility. Positano and Amalfi are the major tourist towns and picturesque but crowded and priced for tourists... the whole coast is picturesque with restaurants aplenty and everyone understands English, so consider a villa outside the two main cities with day trips to them by bus or taxi.
Finally, I always enjoy driving stick shift Euro compact cars on vacation, and did this time in Tuscany, but I'd never want to rent a car on the Amalfi Coast -- the road is too narrow, too treacherous, parking is worse than scarce and very expensive. I liked Vespa-ing in Capri but on Amalfi I'd feel unsafe on two wheels. Ferries link Naples-Sorrento-Capri-Positano-Amalfi to get you staged. You can hire private boats on the coast at many of the towns and beaches, with or without a skipper. The DIY boats have a max 40HP engine, so with 10 people you might be better off getting a skipper with a bigger engine -- anyway it's tough to pilot in the waves and current unless you have experience.
Long answer, but the place was so damn magical I want to share some insights.