All you need to get started is a knife, a stick, a whet stone, plus a couple of bandaids and butterflies.
There are dedicated/designated Whittler pattern knives, and they usually do a very good job, with 2 small blades and one large that are usually designed especially for whittling and carving. Case, Boker, and others make them. Pretty much any knife can be used, though. I like small blades sticking out of decent sized handles for most of my whittling. The most-used blades on my whittlers have always been the pen or small clip. I have also used a Peanut. The blades are the right dimensions for what I like to do, but the handle is a bit small for extended whittling sessions. Works well, though, and is very convenient to slip out of the watchpocket of my jeans and make some shavings.
Wood -- I've been using a lot of cottonwood lately. It is plentiful where I live, and the trees are always shedding branches, so getting some without carrying a saw or axe is simple. Just take the dog for a walk by the river, and I've got plenty of sticks to play with. I also like sugarpine for easy whittling.
I start my cub scouts off with whittling soap. It's even easier than a nice dry piece of cottonwood, and doesn't require much hand strength to put a knife through, so an 8 or 9 year old whose primary daily physical activity consists of moving a joystick around can get used to the movements pretty easily, and then we move to wood. (I aim to raise up a bunch of boys who think that video games are a waste of time, who would rather be in the woods with a pocketknife and a dog.

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Keep whatever knife you're using plenty sharp. If it gets a bit dull and slips, that's when you get to use the bandaids.