Milan has a lot to offer: art, history and shopping. However, it is also rather spread out, and the best way to navigate it is on the subway, which is crowded, hot and uncomfortable. And, it was a little crazy when I was there since the national team had just won a soccer championship.
One of the must-see places is definitely the cathedral, said to be the largest in the world. Once you take a peek inside (be quiet and respectful), take a jaunt up the stairs onto the roof. You can see the gargoyles and spires close up, and also get a bird's eye view of the city. You can take an elevator instead, but that will cost you more, and you'll have to climb more stairs to see some of the best parts anyway.
Another place to check out is the monastery where DaVinci's The Last Supper is located. However, due to the popularity of a certain book by Dan Brown, you now need to make reservations literally months in advance to view the painting.
The train station is also a work of art, with intricate sculptures on the facade. And there is a good deal of public art in the city.
There are tons of hotels and restaurants in all price ranges near the train station, so that is the place to stay. Buy a metro pass and a couple bottles of water and you should be in good shape.