Supplies and splinters.

What supplies, other than plywood, would I need to make a 4x2x2 cage?

Also, would I be using plywood or some other type of wood?

Would I have to coat the wood so the snake can't get splinters?

There's been quite a few threads about this...wood enclosures can mold, warp, etc. If you really want to design your own enclosure, I'd look into using plastic or another non-porous, synthetic material.

What kind of plastic would I use? PVC?

Use melamine

I've heard of that for a rack, would it work as a cage?

Yes, if you are sure to seal all the corners and edges where melamine meets melamine.