6 days in Milano would be too many. 2 to 3 would be more than enough. You can consider various day trips. Verona (of Romeo and Juliet fame), the upper city of Bergamo and one of the lakes (Como or Garda) would be fun. Also, whilst in Milano, try not to miss La Scala, the world famous opera house just round the back of the Galleria. You must go inside to see it.
Bring plenty of films and post your pics to share with us after your stay.
Another word of advice, never give money to beggers. For once you gave to one, a whole queue will follow. And watch out for those gipsy looking mothers (very often carrying a child) with young children. They are very good pickpockets.
I go to Milano for the food -- and for that, or for the company mentioned in an earlier post, six days might not be too many. But the idea of day trips is a good one, and I would add Mantova to the menu -- seat of the Gonzaga, with many things to see and photograph. And a typical cuisine that differs from Milano. It was voted most livable city in Italy. Did I mention the food? And not too far from Bologna -- a great eating city. But maybe you are not going for the food.
Don't know what you consider the usual tourist haunts -- I am always attracted by the roof of the Duomo, from which you get a panoramic view of the city and a huge set of shots to choose from. The Duomo exterior is itself interesting these days, after an extensive cleaning.
Via Dante, leading from the Duomo to the Castello Sforzesco, frequently has interesting photo exhibits -- outside. Don't know what might be showing when you are scheduled there. It is also where you can find New Old Camera, specializing in used cameras, with a heavy load of Leicas. With the current exchange rate, this may be less than enticing, but you can always look, and, on some items, it has recently had prices that parallel US prices.
I agree that part of the attraction of Milan is the street scene -- which in some ways I would put more like New York.
More if you're interested -- but you may not need anything more if you have a local guide.