I buy those big rubbermaid boxes at Target when they go on sale for $4 each. They're clear, I think "sterilite" or something?
I separate by gender and size, and in the box I try to put piles of like things together (onesies, pajamas, pants, shirts, etc.). I keep all shoes separate, all winter and outerwear items separate, and all cloth diapering stuff separate.
I stack them so that the youngest sizes are on the bottom of the pile of boxes, then I can just add a box to the top. I don't label the outside because when I take it down for the next kid, I take out the next size and put in the old size as needed until it's full and completely replaced by the outgrown size (make sense?). I stick a tag with the size so it's visible through the side of the box.
HTH! also makes for easy moving as the tops click on, and I put a sachet or a dryer sheet in to keep things fresher.