The Duomo and the area around it for sure. I'll give you a heads up that there are an absolute ton of ADs selling all kinds of exotic watches in the piazza to the left of the Duomo (as you face it) - if you're going with your other half play dumb and act surprised when you 'accidentally' come across these stores (but bear in mind Gucci, Louis Vuitton etc are also located in very close proximity).
A night at La Scala is very much an event as well. And you aren't far from the lakes so maybe a day trip out there.