This is the place to go when you arrive in Milan to get the feel for just hanging out like a local. People love to stroll around the Duomo area and the inside is amazing to see. Read up on what to look for inside so you don't miss anything. Someone in my group decided to do a confession on the spur of the moment and was welcomed into a confession area where English was spoken. She said it was a good experience so if you are into that it's probably worth considering. Another thing about starting your trip at the Duomo is that it can keep you from getting lost. As long as you know how to get to your hotel from the Duomo (which is probably easy given how many streets converge there) you can always ask a local how to get to the Duomo by saying "Doumo?" anywhere in the city. Just about every trip I did in Milan went through the Duomo.
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