My family (two adults, two kids) just returned from a Wednesday - Sunday (3 full days) trip to Naples. We really had a great time and thought Naples was a fantastic, vibrant city.
Accommodation - We rented an AirBnB on the Spaccanapoli, next to the Quartieri Spagnoli. Our building had an elevator and an fantastic view of the bay and Vesuvius. I don't think we could have picked a better location as the feel was very local, plenty of food options close by and all sights within walking distance.
Transportation - We fly to Naples from Vienna without any issue. We were all checked for fever upon arrival in Naples. Our AirBnb host pre-arranged taxis for us to/from the airport for 25 Euro each trip. We mainly explored Naples on foot, which was not an issue for us at all, but I could see where some less experienced travelers would be intimidated by the local traffic. For our trip to Pompeii, we took the Metro to the main train station and from there took the Circumvesuviana to Pompeii. Buying the tickets for the Metro and the Circumvesuviana were easy. For our trip to Capri, we took the fast SNAV ferry and bought the tickets at the port.
Food - We ate a lot of pizza and had a lot of espresso. Food is everywhere and it is good.
Sights - On Day 1, we had lunch at Di Matteo (pizza) which was excellent. We then joined a local tour of Naples for 2.5 hours which we booked through AirBnB. We followed that tour with the Naples Underground tour, which my kids really liked. Then dinner at Tadem, which is well known for their Ragu. On Day 2, we visited Pompeii for about 5 hours. We started with a guided tour we had pre-booked through AirBnB, followed by a picnic and then more exploring. Dinner that night was at Antica Pizzeria Port'Alba, which we enjoyed. Day 3, we took the ferry to Capri, despite being told by many that Capri in the off-season is dead. In fact, it was absolutely divine as the weather was perfect and there were no tourists. We had lunch at Ristorante Pizzeria Verginiello, which has a fantastic view and was packed. Great seafood. We then walked to the Gardens of Augustus, hiked to Belvedere Tragara, over to the Natural Arch and back to Capri town where we took the local bus down to the marina. On Day 4, we spent the morning in a cafe followed by a stroll through Piazza del Plebiscito where hundreds of small kids were in terrific costumes for Carnevale.
Overall, we loved Naples and will be back to explore more. We loved the friendliness and real vibe of the city.