Kristen, we've been to La Scala a few times; Certainly an incredible experience. The first time we were living in the States and booked on-line through their website. Yes, it sells out quickly and seats are extremely limited - there are many season pass holders and apparently "corporate or embassy seats" always reserved! We got the last two seats in the nosebleed - loved it and swore we'd go back... with better seats.
Two or three years ago we did exactly what Kent suggests and picked up tickets on the day of the performance - totally on a lark. We could not believe seats were available and scrambled to find clothes to wear! We actually had better seats (I guess given our first-run anything had to be better!).
But as Kent says, that was the luck of the draw and NOT something we planned or counted on. If you really want to go, you may have to bite the bullet... WE DID our first time, but again, money well spent!
Ciao,
Ron