I was warned not to go to Naples at all by friends who were concerned about my safety. After all, the city does have some of the highest crime and unemployment in the country. You can even sense it when wandering the streets - no wait, the restlessness is probably just due to the uncomfortable heat of summer.
In truth Naples is fine within the confines of the tourist area, which is basically the triangle between the train station in the east, the Archaeological Museum in the west and the Royal Palace in the south. Even so there's a bit of a gritty feel to it; the one time I was out alone after dark I was a bit paranoid but I kept to the busy main streets. Having just arrived in Rome last night the contrast is quite stark. (Polizia! Everywhere! As compared to not a single one seen in Naples).
Honestly I didn't spend a whole lot of time in Naples; my first day I went out to visit the famed sites of Pompeii and Herculaneum, and the second was spent on a boat circling the island of Capri. But it was definitely worth wandering the city's historical center, which has amazing classical architecture, beautiful churches and lots of cool (albeit touristy) shops. Oh, and the pizza! Naples is the birthplace of pizza so I made sure to try out a few different restaurants.
Every time I've moved in the past several years, I find the local drivers to be increasingly worse and more ruthless than the last. (I've previously lived in the Bay Area, New York, and now Paris). But Italy takes the cake on aggressive drivers with no regard for pedestrians. To cross the street one pretty much has to stay on the heels of a local, otherwise risk being stranded on the sidewalk for a very long time. If there aren't any around, though, you just have to take a deep breath, step out onto the street and stare down whatever car or scooter is speeding unrelentingly toward you. If you do go Naples, use Hostel of the Sun as your home base. The staff are incredibly friendly and knowledgeable, and you're sure to meet some cool people who can become your new friends and travel buddies.