Besides what are mentioned, depending on your interests, a few more places:
1. Pinacoteca di Brera (lots of masterpieces)
2. Ambrosian Library (has Da Vinci's painting and his drawings)
3. When visit the Duomo, don't forget the Duomo museum where many original artifacts are displayed.
4. Of course, Santa Maria delle Grazie for the Last Supper (reservation required).
It may take 2 - 4 days in total.