Milan is one of the world capitals of fashion, so, of shopping too.
The shopping itinerary starts from the Duomo (Cathedral), and many shops are nearby, in the following streets:
Corso and Gallery Vittorio Emanuele (with The Highline Outlet), San Babila Square, Via Torino, Via Manzoni.
From San Babila square, you can reach, on the left, the "Quadrilatero della Moda" (fashion district), four small streets (Via Montenapoleone, Via Sant'Andrea, Via della Spiga, Via del Gesù) hearth of the luxury boutiques.
On the right you can reach Corso Venezia and Corso Buenos Aires, kilometers of shops of all types for all people.
On the opposite, from Via Torino you can reach Porta Ticinese, the area of alternative shops, very close to two of the night life disticts: The Navigli and San Lorenzo Columns (link).
Outside the city center many commercial areas are available, like CityLife Shopping District in the homonymous modern skycrapers area (link), and many others Outlets.
Outlets near Milan:
Scalo Milano – Outlet & More
Via Milano 5 – 20085 Locate di Triulzi (MI)
Vicolungo The Style Outlets
Piazza Santa Caterina 1 – 28060 Vicolungo (NO)
https://vicolungo.thestyleoutlets.it/en/home
Segrate Outlet Village
Via Cassanese 77 – 20128 Segrate (MI)
http://www.segrateoutletvillage.com