Just to further add to this: Summer is indeed a very busy period for all beach/sea destinations, particularly in the south as the weather gets “beach” hot from around May and allow swimming / beach visiting until almost October. The really busy period is mid July to end of August, and generally all week ends from May to October, but week days late June /early July should be pretty quiet.

With regards to natural beaches crowding, the problem with the Sorrento /Amalfi Coast is that it is mainly rocky and the beaches are rare, so some private sea access is done via platforms rather then sandy beaches, but by Italian Low a percentage of any natural sandy beach needs to be public. Also private clubs/ “lido’s” need to allow sea access without payment but obviously if you need a parasol or some Sun chairs/beds you need to pay for it .

if a sea vacation is the main purpose, then Positano/Amalfi coast may be second to none, albeit Positano itself is quite expensive as a base I would think. The main luxury hotels there are really world class though ( San Pietro and Le Sirenuse). As the previous poster has stated there are also plenty of options From Sorrento all the way to Cetara for beach/sea spot. Personally I would however choose Nerano as base for a longer and then visit the other towns including Positano.

Monastero Santa Rosa Hotel (closer to Amalfi) may also be one of the most stunning location hotels in the world (particularly impressive the infinity pool).

with regards to where to stay in Naples, see my posts in this thread Naples hotels

with regards to what to do in Naples, considering that the whole of the historic centre has been a UNESCO protected site for years, I would be happy to contribute in a dedicated thread but also suggest to see the BBC’s programmes “Italy’s Invisible Cities” Ep. 1, and “Italy Unpacked” series 2 ep. 3. If at all possible.

in no particular order, things that must be seen, are San Severo Chapel with the unbelievable “Veiled Christ sculpture”, San Lorenzo Church with the Greek underground city remains, Santa Chiara, San Domenico, Naples’s Cathedral with San Gennaro museum and jewels (valued higher that the a British Crown Jewels), and the almost unknown San Giovanni at Carbonara church (heavy on the church side but unbelievable example of Gothic, Baroque and Renaissance architecture). The aristocratic palaces on Via Dei Tribunali and Via Toledo, the Royal Palace (including the Gardens), San Martino (with both Sant Elmo Castle and the abbey), Castel dell’Ovo, Borgo Marinari, and the three main museum of National Archeological Museum, Capodimonte and PAN.

The above is really, really a condensed list, omitting also short trips to the Likes of Pozzuoli, Cuma or the islands of Ischia and Procida.