Go to the city center and check out the Duomo - it's gorgeous and free to go inside the first floor. From there walk straight back from the Duomo and bear to the right of McD's and go straight back. 10 minutes later you'll be at Castello Szforsco, fat ass castle that's old as fuck, and behind that is Parco Sempione where the Albanians sell hash.
The Navigli area is also a really cool spot with great bars and aperitivo on a couple canals.
Bunch of good clubs around as well if you're into that sort of thing.
Try and go see the the Last Supper if you have a student pass. Otherwise I think it's pretty spendy.