This is a tough one. It is a city of extremes and contrasts. This can be seen from shopping in a street market in one neighborhood to a street filled with high end labels (Gucci, Prada, Armani) in another. The old town can feel almost claustrophobic with its narrow streets but then by taking the funicular up the hill you have tree lined streets and sweeping views of the bay. One cliff side area has grand villas while another area has clothes drying from the balconies.
It is true what they say about the pizza of Naples; it was truly the best. Its all in the crust, well and maybe the tomatoes which are the sweetest you will ever taste. You'll also find it hard to beat the seafood; octopus, mussels, clams and calamari.
You don't need to shy away from Naples, every big city has its issues. It doesn't have the art of Florence, or the history of Rome, but somehow it slowly finds it way into your heart. It may be because, with fewer tourists, you feel like you've had more of a true Italian experience.