In my opinion the most beautiful old building in this rather unlovely city is the pink-stoned Duomo in the central square. The top floor cafe in the neighbouring Rinascente department store (itself a heavenly paradise, albeit for shopping) has fantastic views of the higher levels of the cathedral with the gargoyles looking you right in the eye as you drink your coffee.
Luini (Elementary B), 16 via S.Radegonda, down the side of Rinascente.
Perhaps the most famous bakery in Milan, since 1940, this is the best place to try panzerotti (deep fried pizza dough filled with mozzarella, garlic and tomatoes) as well as bread and other hand-made specialities. A panzerotto costs about €3 and a coke €1.50.
Pizzeria di Gennaro (Intermediate B), 16 via S.Radegonda, next to Luini.
Milan’s oldest pizzeria, not particularly special but friendly efficient service and the food is fine. I had a margherita for €6 which was ok. A good place to rest your feet after shopping in Rinascente.
For followers of fashion, the nearby district of Monte Napoleone, at the end of via Buenos Aires, is the principal shopping district. One of my favourites is Missoni on Via Sant ‘Andrea.
If you like Alessi designs, you should visit their stores at 9 Corso Matteotti and 14 Via Manzoni.