As for what to actually do your cutting with, I use the same old linoleum cutting tools I've had for years. Speedball makes a nice set; you get several different tips that conveniently store in the handle; you can pick this up online or in a lot of different craft or art supply stores for less than twenty bucks. If you want to go even cheaper though, a plain old hobby knife or Exacto would work, too.
So. You've got this
Mine's just a little chunk of plasticy stamp carving material I picked up for cheap, but it has a nice consistency. I find it helpful to use the knife edge and go around the outline of my design before I do any real removal of the block. This will help keep your lines cleaner and smoother in the final product.
There's not really an particular way you should cut, unless you're going for a particular look. I really like rough cuts with a little undercutting, so I use the broadest tooltip that I have and clear out big chunks at a time, working around the image and avoiding going "inward" into it for the time being. In this case, inward being the wolf's mouth, where I wouldn't have room to maneuver that wider bit around.
Yeah, don't be a dumbass like me and choose your work surface carefully, or take some means to protect it, like with a cutting mat. Thankfully my computer desk is a cheap piece of particle board crap that I've had for way too long and I don't care anyway. You should take care with any sharp tool though, and whenever possible, cut away from yourself, not towards. I've got a scar down my left index finger from a linocutting project gone awry.
Time to move onto smaller tips and hash out some details. You should always start with a light stroke and increase the pressure, especially when moving from a narrow area to a wider one; you can always go over a cut again, but it's kind of hard to fix things if your hand slips and you cut away more than you meant to. The consistency of your carving material really comes into play here. I really like the brushy swipes that the wolf's head ends in, so I'm really trying to preserve that with thinner cuts.
If at any point in your carving you're not sure how close to being done you are, there's a great test: tap that bad boy on some ink and try it out on a piece of scrap paper. The above impression is stamped with Tim Holtz's Distress Ink in Fired Brick, next to one of the photocopies I messed up the first go-round. If you're like me and you like undercutting, you can leave all those messy little details, and you can see how some of your shallower cuts that are harder to read when everything's the same color (here I was checking on his eye) are shaping up.
Chances are very likely that your final image won't turn out exactly like your original drawing or print, and that's okay. I personally feel there's a certain beauty in the effects that you get out of the carving process itself. I really like the more angular look that the stamp has, and the dirty grunginess that all the little extra bits that didn't get taken out give it, so this is where I'm stopping at this particular piece.
Here you see my stamp mounted to an old CD; the brand of carving material that I use has a sticky back for temporary adhesions like this, but you may want to mount your stamp more permanently on a wooden block or something else more comfortable for you to grip.