Museo del Novecento: Directly across from the Duomo sits this contemporary art museum in a great contemporary building. The views from the top floor of the Duomo are really amazing. A large work in neon is featured at the top which is a sharp contrast to the architecture of the Duomo below. Featuring Italian artists, the collections begin with works in the early twentieth century and end with works form 1968. You will see works by Morandi, Burri, Licini and other artists I had not seen before. There are a few non-Italian artists like Picasso featured at the very start. Another plus is that it is open on Monday afternoons until the early evening, so it is open while many of the other museums are closed for the day.
Museo Poldi Pezzoli: At Via Ugo Foscolo 3, this is a museum featuring older Italian works of art in a beautiful palace. Here you will see renaissance works by Botticelli and others, some amazing stained glass and a really wonderful collection of armor. I am not normally a big fan of armor collections but this was exceptional in how it was displayed.
Santa Maria delle Grazie: This is the home of The Last Supper by Leonardo da Vinci. It is one of the most famous paintings in the world. It is essential that you book tickets before you arrive. See if your hotel concierge can do it for you or go online to Tickitaly which is one site you can use.