In gradeschool, I remember being given a rubbery block of linoleum and a sharp (very sharp) carving blade with which to create my design. I carved a brontosaurus and remember that more than half of the kids in class ended up cutting themselves in one way or another. About six months ago I purchased a Speedball Block Carving kit at Hobby Lobby which sat unused on a shelf in the corner of my shop -- until yesterday. On a whim I decided to give it a shot with a basic carving just to see how it worked.
I found that re-inking the block each time produced the best results.
And last but not least: I stamped the inside of a "secret compartment book" that I finished last week. Unfortunately the cover didn't turn out so well, so I ended up just painting it black.