The following are some suggested activities and sights for certain kinds of visitors:
The following is a tried-and-true route through the historic center; you can reverse the order, skip steps, etc. as your schedule allows. See the Naples sights page for details on the various stops.
Start at the Montesanto metro: walk through Pignasecca Market (8-1 very "Neapolitan" fruit/vegetable market, more on Markets page) to Via Toledo
At Via Toledo, sample Gelateria della Scimmia, then head north...then turn right onto Spaccanapoli
Possible detour: instead of heading straight north, you could go south to Caffe Gambrinus, Galleria Umberto, the Royal Palace, and/or the San Carlo opera house.
Possible stops on Spaccanapoli: Gesu Nuovo, Santa Chiara, Sansevero Chapel, and Scaturcchio...also a second chance for Gelateria della Scimmia. Piazza Bellini, a slight detour off Spaccanapoli, is also a nice spot to stop and relax (and check out the ruins of a Greek wall).
Walk up Christmas Alley; check out the Limonè limoncello factory/shop (maybe have some granita!)
Possible stops: Underground Tour, various other churches
Head back to Spaccanapoli, and walk to Pizzeria da Michele (and maybe Gelateria Polo Nord)
Wander back to the Piazza Cavour metro to head home
Possible stops: Duomo, Archeological Museum
For a longer walk:
Instead of starting at the Montesanto metro, go to the Piazza Amadeo metro: walk through Santa Lucia district (and/or along the water) to Piazza Plebiscito and Galleria Umberto; then walk north on Via Toleo to get to the Centro Storico.
More stops/detours on this longer route: Castel d'Ovo...or head up funicular to Vomero for Castel Sant'Elmo and/or Certosa San Martimo
The most famous sites in Naples—best for people staying just a short time
Eat at Pizzeria da Michele (or another top-notch pizzeria)
Walk around the Historic Center (see above)
Ferry to Capri (or other islands), or ferry/drive to Amalfi Coast (or Sorrento)
Best sites for people who love to eat
Check out Naples restaurants - pizzerias, seafood pasta, etc.
Get sfogliatelle at Attanasio's, baba at Scaturchio, and coffee at Caffe Mexico (see to eat page for more)
Mozzarella di Buffala at Vanulo Creamery
Gelato at various gelaterie
Go to a fish/produce market
Check out a food festival (see Events page for listings/resources)
Itinerary for people who love the outdoors
Hike the Amalfi Coast (e.g. Walk of the Gods)
Capri (hiking, Blue Grotto)
Hike Mount Epomeo on Ischia
Explore the (relatively) undeveloped coast of southern Campania (e.g. Cilento National Park near Paestum)
Naples is a treasure trove of ancient, medieval, renaissance, and modern history
Naples Underground Tour
Archaeological Museum
Castles and Palaces (Castel del'Ovo, Castel Nuovo, Castel Saint Elmo, Royal Palace at Caserta, Royal Palace at Plebiscito, Castello Arogonese on Ischia)
Campi Flegrei sites:
Piscina Mirabilis at Miseno
Pozzuoli Tempio di Serapide, Amphitheater, and Rione Terra
Cumae Greek Sybil Ruins and Acropolis
Baia Archeological Park/Ancient Roman Baths
Castello Arogonese in Baia including its Archeological Museum
Grotta di Seiano (ancient Roman tunnel under Posilipo to Roman ruins on the coast)
Paestum Greek Temple Ruins
Villa Jovis on Capri
For those who want to chill and take it easy
Passegiatta along the Lungomare from Mergellina to Piazza del Plebiscito
Chill on the beach (e.g. along Amalfi Coast: cheap public beaches include Atrani near Amalfi, or Fornillo near Positano)
Lounge about in Capri
Go to a spa on Ischia
Cafes and restaurants
Hang out in a hill town (e.g. Casertavecchia, Sant'Agata dei Goti)
People watching (e.g. in Piazzas, such as Plebiscito)
Caserta's Royal Palace appeared in Star Wars (where it plays the palace of Naboo), Angels & Demons, Mission Impossible III, and Conclave (in several movies it stands in for the Vatican)
Ischia appeared in The Talented Mr. Ripley and parts of the TV show My Brilliant Friend
Procida was the backdrop for most of Il Postino
Positano served as a romantic getaway in Under the Tuscan Sun (and appeared, along with Capri, in the 2019 Bollywood film War)
Pizzeria da Michele in Naples was in Eat Pray Love
The Sopranos filmed the episode "Commendatore" along the waterfront (near Castel dell'Ovo), in the Centro Storico (on via Tribunali), Bacoli/Monte di Procida, and Cuma (at the Grotta della Sibila)
CampaniaMovieTour has a Google map of movie sites in Naples
Want to party all night long? Here are some of the hotspots:
Mergellina/Via Chiaia - nearly every night, this area becomes inundated with young people looking for a good time
Via Napoli in Pozzuoli - a string of bars for every style, close to downtown Pozzuoli; especiailly popular in the summer
Capri - especially in the summer, this enchanted island becomes party central; you may run into celebrities at Number 2...also recommended by partyearth.com (a nice resource to party spots) is The Underground in Anacapri.
...also check out the Nightlife page
Don't have much money? Here are some free (or close to free) fun things to do in Naples:
Centro Storico Walk - (see above) many of the churches and other sites are free; plus, the neighborhood itself is the draw
Metro & funicular - for a mere €1.10, you can easily get to Vomero for a stunning view of the city
Da Michele Pizzeria - at only €3-4 a pizza, you can have a slice of heaven without breaking the bank
Pozzuoli Tempio di Serapide - has no charge to see, plus you can walk around the Pozzuoli port area and check out the fish market
Naples has a number of art works tucked away in odd places...
Churches downtown (such as Pio Monte della Misericordia which has Caravaggio's "Seven Acts of Mercy" and Capella San Severo with its amazing sculptures)
MADRE and PAN museums
Line 1 Subway stations
Royal Palace at Piazza del Plebiscito
Capodimonte Museum (not the most thrilling collection, but has a wonderful Caravaggios)
Do you love things that are a little bit scary and dark, and maybe haunted? Here are some things to check out (see Naples Sights page for details):
Naples Underground (Napoli Sotterranea) tour
Purgatorio ed Arco church (with its skull artwork, hypogeum, and shrines to the dead)
Sansevero chapel (especially the creepy preserved bodies in the room next to it)
San Gennaro and San Gaudioso catacombs
Fontanella cemetery --> is this open??
Tombe dei Poeti/Crypta Napoletana (Virgil's tomb and ancient Roman tunnel)
Grotta di Seiano (another ancient Roman tunnel)
In the 17th through 19th Centuries, young British aristocrats toured Europe following what was called the "Grand Tour," much of which was in the Naples region. The tour focused on art and history, including the many connections the region has to ancient literature. You can re-trace their steps, both for the fun of it, and perhaps because you love ancient mythology and literature yourself:
Lago Averno (inspiration for Virgil and Dante)
Miseno (site of the ancient Roman fleet)
Pompeii
Herculaneum
Vesuvius (it was more exciting for the Grand Tour people because it was erupting for most of the 18th century...)
Paestum
Capri Blue Grotto
Have a caffee at Caffee Gambrius (Piazza del Plebiscito)
For a Grand Tour ambiance, check out the nicest hotels on the Lungomarea and Posillipo (such as Grand Hotel Parker's or Vesuvio), and the San Carlo opera house...or Sant'Agnello near Sorrento