Milan is the business and fashion centre of Italy. For many, it is just the place to come to shop till you drop.
If you expect a beautiful renaissance city like Florence, you would be disappointed. Although there are some Roman remnants and many beautiful medieval and neoclassical buildings, Milan is on the whole a modern city, partly due to the heavy bombings suffered during World War 2.
The Cathedral (Duomo), the Last Supper (for which an on line reservation is a must), the Castle (Castello Sforzesco) and the Pinacoteca di Brera (Art Museum) are the absolute must-see monuments.
Beacuse of its location and easy public transport links, Milan can be used as a base to see the rest of the North of Italy, notably the lake district. Lakes Como, Maggiore, Lugano and Garda are all easy day trips from Milan. Also very near to Milan is the Ligurian riviera, a one day trip from Milan could very well be Portofino, where the seriously rich go for seeing the other seriously rich.
Also, if you are a soccer enthusiast, Milan is home to two of the strongest italian football teams, Inter and Milan. Both play at the San Siro stadium, the largest in Italy.