Chiaia. When Neapolitans need a break from the crowded centro, they head to upscale, seaside Chiaia, one of the city's most affluent neighborhoods.
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Sorrento is more touristy than Naples and is known for its lemon groves and limoncello, charming cafes, and access to the Amalfi coast. Naples is well known for its colorful atmosphere, long running history, and influence on world cuisine as the birthplace of pizza. Also, Naples is famous for Piazza del Plebiscito.
Forcella, Rione Sanita`, and Secondigliano also have reputations although these areas are probably more known for organized crime hits/dealings. Petty crime like getting robbed can happen anywhere in Napoli because the thieves go everywhere, including the nice areas too.
I would say if you are all looking for lots to do - and a big city hustle & bustle atmosphere, then Naples may be best. Or if you are looking for relaxation/swimming/generally chilling out in a beautiful seaside village, then Positano or elsewhere on the Amalfi Coast may be best.
Naples, with its ancient streets of Neapolis and Quartieri Spagnoli, is a walkable city; you can explore everything from the royal palace of Naples to Castel Nuovo on foot.
We recommend spending at least three days and two nights in Naples. This will give you enough time to enjoy the city at a relaxed pace. It gets very hot in Naples during the summer months, so remember to account for this when planning how much you want to do each day.