Palace of Caserta

The Palace of Caserta was one of the largest Royal residences built in the XVIII century, comparable only to Versailles, though the two residencies are different for architecture and style.
The Palce of Caserta is a Unesco World Heritage site.
There is also a 12O Hectares park with majestic fountains and hidden places.

It's possible to get there by train from Naples

http://en.wikipedia.org/wiki/Palace_of_Caserta

Pompei and Ercolano

Pompei and Ercolano are two of the most famous Italian touristic sights. As you might know, the cities were destroyed and buried by ash by the Vesuvio eruption during the Roman Empire and stayed untouched until they were discovered and restored in the XVIII century. These ruins, as big as a city, offer a great overview over roman life, with wonderful mosaics.

It seems it's possible to visit them by train with the Circumvesuviana, a suburban train with few lines going up or around the volcano Vesuvio.
Pompei is half-way to Sorrento. By train, from Naples to the coast of Sorrento is just one hour! In the sunny days the view on the Gulf of Naples and Vesuvio is amazing!

Historic center

Naples' historic center is Unesco's World Heritage, and it's considered the largest in Europe. It's a mixture of styles: medieval, renaissance and baroque. The best way to visit it is just strolling aimlessly, in order to enjoy the alive and exciting center full of people shouting and speaking like Enzo! You'll see that Naples can be frenetic and slow at the same time.

Some parts of the old center are considered dangerous especially at night, above all Quartieri spagnoli, but during the day two poor travellers like us shouldn't have any problem....

http://en.wikipedia.org/wiki/Quartieri_Spagnoli

Spaccanapoli

Spaccanapoli is the most central street of Naples, the one is cutting the city in the middle. It's considered the best place to experience the real lifestyle in the city: food smell from the windows, black market in the street, little corner shop selling amazing things, people shouting and motorini speeding up: Naples!