Milan is the Italian city people write off before they arrive. It has no Colosseum, no Venetian canals, and a reputation for fashion, banks and traffic. Then you start pulling the thread. It was a capital of the northern half of the late Roman Empire, which is why some of its churches go back to the 4th century. The Visconti and the Sforza turned it into a court city, Leonardo painted The Last Supper here for Ludovico Sforza in the 1490s, and in 1805 Napoleon had himself crowned King of Italy inside the Duomo. Allied bombing in 1943 hit the city hard, and the center is a patchwork of eras because of it.
The city rewards planning more than wandering. The three things everyone wants, the Last Supper, the Duomo roof and an evening at La Scala, all need tickets before you arrive. The rest is rhythm: a coffee at the bar, a long museum morning, and the aperitivo, a drink and snacks between 6 and 9 pm that is the real start of the evening. Three days gets you the headline sights at a run. Five lets you add Brera, the Navigli, the skyline of Porta Nuova and a day trip without hurrying.
The plan keeps each day in one part of town: the Duomo, Galleria and La Scala; the Last Supper, the Castle and Brera at night; the Pinacoteca and Porta Nuova; the old churches and the Navigli; and a day trip to Bergamo. Start on a Tuesday, because the Last Supper, the Castle museums and the Pinacoteca are all closed on Mondays. Milan is also the easiest door into Italy from the north, and the same trains reach Lake Como, Lugano, Turin, Genoa, Verona and Venice.