The Godfather trilogy follows the Corleones, a New York Mafia family whose roots are in the Sicilian village of Corleone. Coppola filmed the Sicilian scenes in real villages near Taormina, and most of them can still be visited; the New York and other American places come after Sicily. Start in Savoca at Bar Vitelli, where Michael asks to meet Apollonia, then walk up to the church of San Nicolò, where they marry.
Savoca and Forza d'Agrò, which plays Corleone, fill half a day. In Forza d'Agrò see the Mother Church and its steps, the Holy Trinity square and its arch and the house where Vito was born. On another half day, visit the Castello degli Schiavi, Don Tommasino's villa, by reservation; the Taormina-Giardini station is on the way back, and you can add Motta Camastra above the Alcantara valley or the Pennisi di Floristella castle in Acireale.
Part III ends in Palermo, a separate trip of about 2h30: the steps of the Teatro Massimo and Villa Malfitano, with the temple of Segesta and the church of Sant'Orsola in Erice further west. The Sparagogna station is a disused stop in open country. Check before you go: the castle and several houses are private, and many churches open at irregular times.
In New York, the Corleones' city, start in Little Italy, where everything is a few minutes apart: Old St. Patrick's Cathedral, where the baptism of the first film and Michael's honour in Part III were filmed, the Genco building on Mott Street, where Vito is shot, the Mulberry Street Bar and Elizabeth Street; then walk to East 6th Street, the young Vito's street in Part II. Downtown, add the courthouse steps and the Custom House; in Midtown, Radio City Music Hall and the St. Regis. Calvary Cemetery in Queens, the Corleone mansion on Staten Island and the site of Louis' Restaurant in the Bronx are further out.
Further away are Fleur du Lac on Lake Tahoe, the family compound of Part II, best seen from a boat, and Beverly House in Beverly Hills, the producer's mansion of the horse's head, seen only from the street. In Santo Domingo, which stands in for Havana in Part II, see the El Embajador hotel, the National Palace, the Agua y Luz theatre and the National Library. In the Plan section you choose which part to plan: Sicily, New York, Lake Tahoe, Los Angeles or Santo Domingo.