As I acquire them, I toss the boxes into the garage.
On Garbage Eve, I break them down (if I have a lot of it, I'll actually cut all my cardboard into ~16" squares so it stacks neatly in the recycling bin - I can get a lot more boxes in there that way, which is useful after an amazon binge or a physically large purchase) and put them, along with any recyclable packing materials, into the recycling bin. Everything else goes in a black garbage bag and gets put out with the trash bin.