1) Santa Maria presso San Satiro (free). Built by Bramante, this hidden church features the world's first example of illusionistic perspective in architecture.
Via Torino 17-19. Metro Duomo (M1 - M3).

2) Ca’ granda (free). The former general hospital of Milan, it is a masterpiece of Lombardy's 15th century architecture, by architect Filarete. Now it houses part of the State University of Milan.
Via Festa del Perdono. Metro Missori (M3).

3) Santo Stefano Maggiore church. You can have a quick look at the facade of this church where Caravaggio was baptised and the Duke Galeazzo Sforza was assassinated.
Piazza Santo Stefano. Metro Duomo (M1 - M3).

4) Museo Bagatti-Valsecchi. A historic house museum displaying a 15th century lombard palazzo.
Via Gesù 5. Metro Montenapoleone (M3).

5) Museo Poldi Pezzoli. It houses a magnificent collection including pieces by Botticelli and Raffaello, and ancient suits of armour.
Via Manzoni 12. Metro Montenapoleone (M3).

6) Castello Sforzesco (partly free). One of the largest castles in Europe, this Medieval complex located right in the heart of the city now houses several exhibitions. Most relevant are the Sala delle Asse with frescos by Da Vinci (it is part of the Museo d'Arte Antica) and the Pietà Rondanini, last work of Michelangelo Buonarroti.
Metro Cairoli (M1).

7) Santa Maria delle Grazie church (free). It was partly designed by Bramante. It also houses Leonardo Da Vinci's Last Supper (for the Last Supper it is usually necessary to book weeks or months in advance).
Piazza di Santa Maria delle Grazie. Metro Conciliazione (M1).

8) La Vigna di Leonardo. A recently opened museum, displaying for the first time both Palazzo delle Stelline and Casa degli Atellani, where Leonardo Da Vinci's vineyard was located. Advance booking is suggested.
Corso Magenta 65. Metro Cadorna (M1 - M2) / S.Ambrogio (M2)

9) San Maurizio al Monastero Maggiore church (free). The so-called "Sistine Chapel of Milan" was entirely decorated by Da Vinci's pupils.
Corso Magenta 15. Metro Cadorna (M1 - M2).

10) Chiostri di Sant’Ambrogio (free). Designed by Bramante, they now house the Università Cattolica.
Largo Agostino Gemelli 1. Metro S.Ambrogio (M2).

11) Pinacoteca e Biblioteca Ambrosiana. The Ambrosiana gallery and library includes, among many other pieces, Da Vinci's Atlantic Code and his "Ritratto di musico".
Piazza Pio XI. Metro Duomo (M1 - M3).

12) Cappella Portinari in Sant’Eustorgio (free). The Portinari Chapel within Sant'Eustorgio Basilica is a fine example of Florentine Renaissance.
Piazza Sant'Eustorgio. Metro Porta Genova (M2).

13) Navigli (free). Milan's canals are the highest engineering achievement of Leonardo Da Vinci, who invented the gate mechanism for sluices. The most prominent example is Conca dell'Incoronata in Via San Marco.
Metro Porta Genova (M2).