Voidless plywood. Looks better and lasts longer. For the drawer bodies, 1/2 inch plywood. 3/8 inch plywood if the drawers are rlitively small. Any thinner than that, and screws will not work, so the drawers would need to be framed. For the top, i would use 3/4 inch ply, again void less, unless the plywood is supported every foot, then you can use thinner ply.
Glue everything! The strength is from the glue. Screws and/or nails really just hold everything together until the glue sets.
If you plan on painting the boxes, Kraft faced plywood looks quite professional, or alternatively monstaliner would do a great job of hiding all your little mistakes.