For targets, I use 1/2 plywood boxes, about and inch and half deep, lined with carpet and rags, with cardboard facing, under the paper targets.
This is pretty much what I use, too. Works fine, and carpet scraps are free. The plywood is probably overkill, but I think cardboard is a little low-rent -- my nice air guns deserve better.
I wash my hands well when I'm finished, but other than that, I don't worry about it.