You're likely slowly adjusting to the contrast in sound between the noise of the square and the quiet inside. But imagine the great sounds of a solemn Mass, when the Cathedral is filled with the faithful and resounds with songs and organ music!
Speaking of celebrations: in certain periods of the year, the so-called "Quadroni of Saint Carlo" are displayed between one pillar and another: they are a series of large canvases depicting scenes from the life and miracles of Carlo Borromeo, and are paintings that have been completed by various Milanese masters from the seventeenth century on. If you are visiting during one of these periods, the Cathedral will seem like an art gallery.
If you go under the main altar, you can also see the place where the saint is buried.
In fact, it was St. Carlo himself who, among other things, wanted the famous floor which extends over the Cathedral's entire surface. It is inlaid with three different colors of marble: white, black, and red. If you look down, it's almost like walking on a stone carpet!
Now you can start to go around the Cathedral, and as the oldest and most "authentic part" is at the back, note that if you enter from the front you will go back in time as you proceed farther back.
After just a few feet, note the sundial on the floor that was traced there in the late eighteenth century by Brera astronomers. It is 24 meters long, and presents the symbols of the zodiac. At the back, high above the main altar, you can see a small red light: that's where a gilded copper box is located that holds the Holy Nail of the Cross, a gift from Emperor Constantine. Once a year, the archbishop of Milan goes all the way up there to pull out the box and celebrate a solemn mass.
You can also go see the Cathedral's first stone, which was laid in the year 1386: it's wedged into the right wall. As the church was founded by the Duke of Milan, Gian Galeazzo Visconti, a radiant sun can be found at the top of the central window, which is in fact the symbol of the Visconti family.
FUN FACT: the Holy Nail is the Cathedral's most precious relic. Legend says that it was found by St. Helena, and her son Emperor Constantine used the Holy Nail as a bit for his horse.