Pompei can take a day at least to get around, it's fascinating and will not last much longer, the way it's being neglected.
I heard a few years ago they may close it to the public due to the pounding it is taking, which prompted our visit a couple of years ago.
As PP have said, there isn't much beach in Sorrento but lots of the larger hotels have pools and some of the cliff top ones also have pool + lift down through the cliffs to a bathing deck at the foot of the cliffs. This is not as alarming as it sounds, as they have created artificial rock barriers (correct word escapes me) which take the brunt of the waves, so you have a sheltered sea area for swimming and snorkelling - there are plenty of small fish if you do take a snorkel and mask.
We stayed in Sorrento and found it very touristy and crowded, though we were there in high summer (my God the heat!!!). I would try to find a self catered apartment with air con, as eating out can also be very expensive. We stayed in the Rota Suites and had the use of the pool and bathing platform at the sister cliff top hotel 5 mins round the corner. The Rota Suites were good, actually, and a couple of UK tour operators offer packages with them included. We were on the front, looking out to sea/across a lemon grove - the views were fantastic, as were the sunsets, but it was noisy when all the Apes (little three wheeler vans) and mopeds were thundering past. It was OK with the windows shut, though, as the double glazing was very effective. I've heard some of the little villages around Sorrento are nicer (e.g. St Agnello) and some of the hotels in satellite villages have a free shuttle service into Sorrento.
Remortgage your house if you decide to go to Capri for the day as it is even more expensive than Sorrento - or take a packed lunch and drinks to keep costs down. IIRC it was about €120 for day return fares for 2 adults and 2 children under 10 to Capri on the fast ferry/hydrofoil. There's not that much to see in the town you land at, although I've heard there is some lovely hiking countryside on the island - tricky to do in a day though.