I've already built and had running a system of four cages for my corns and am in the process of building stackable cages for my BPs. So far, my methods and materials have worked just fine.

I use Exterior (roof/wall) top grade plywood 3/4" for my cages, then paint them using Rust-Oleum Latex based paints. These paints are very water resistant if not waterproof. To give extra coverage, I use either sturdy contact paper or clear vinyl on the inside of each panel and tile or vinyl for the flooring.

Stay away from Pine, Cedar, and pretreated woods as those can be toxic.