Naples rewards patient wandering more than any specific checklist. The historic centre is one of the densest concentrations of medieval churches, baroque chapels, and hidden squares in Europe, stitched together by the ancient Roman street of Spaccanapoli. The single essential museum is the Museo Archeologico Nazionale, which holds the greatest collection of Roman antiquities in the world — most of it excavated from Pompeii and Herculaneum. The Cappella Sansevero contains the Veiled Christ, one of the most astonishing sculptures ever carved in marble.
For geography and atmosphere, walk down to the waterfront at Castel dell'Ovo, then up to Piazza del Plebiscito and the Palazzo Reale. Out of town, the Phlegraean Fields west of the city are a supervolcanic landscape with Roman ruins, sunken cities, and the oracle at Cumae — less visited than Pompeii but every bit as extraordinary.