Not much, really. You need a good pencil, decent paper, and a few good erasers. That's it. One thing I should drive home is that buying great art supplies; or in fact, buying things specially set aside as art supplies, will not make you better at art any more than buying a better football will make you a good football player. You simply need something that is functional - to extend the football metaphor, it would be worth getting a better ball if yours had a leak and was always a bit flat, or if yours had some funny lump in it; but any brand new ball from a local store would be fine. With these things, there is a point of extremely diminishing returns. If you spend more than $25 on the sum total of one batch of these art supplies, you are wasting your money.
I do all of my drawings with a mechanical pencil, bought at my local Target™ store. Cost me about $3. I've grown quite fond of a specific make - the Pentel e-Sharp series with its "Lead-Maximizer" technology, and have grown so because that specific brand has a slightly unusual tip, which holds the lead more tightly throughout the life of a single stick of graphite. Generally, towards the end of a stick of graphite in a mechanical pencil (especially the cheaper "Bic" variety), the stick will no longer be gripped by the lead advancer inside the pencil. This allows it to rotate freely, and also slide in and out of the pencil (potentially falling out, if you held the pencil in the open air, pointing straight down). This causes a loss of certain fine control, and sometimes causes accidental strokes when the pencil is close to the page and the lead falls down and onto the surface of the page. A good mechanical pencil like mine solves this problem.
Certain professional artists will get these things titled "lead/graphite holders" - not devices that advance the lead for you, but ones whose head simply grips a stick of lead like a vise and have a pencil-like outer shell for you to grasp and draw with. These solve the same problem, though I've never used them (and actually have never seen on - the above info I give is as a secondary source, and might possibly be inaccurate). What I've got works, and I haven't gotten around to experimenting with that yet.
I use a .5mm lead pencil. .3mm pencils can be found for purchase, though they are considerably harder to find than the .5mm and .7mm pencils available at your local corner store. When I need something sharper than the .5mm width point, I take a very sharp piece of metal (a razor blade works well), and hone the tip down for maximum precision.
Paper, like ice for a skater, can actually mess with your work if it's too rough. You (like myself) can get away with using nothing but mere printer paper, rather than buying fancy art paper. Just be warned that while some works beautifully, certain varieties aren't so reliable. You want to get paper that is very bright (probably about 95 or so, at least by HammerMill's rating system), and paper that has an extremely smooth surface, which you can only judge by trying it. An excellent way to do trial and error is to swipe a few sheets from any copiers you encounter; usually they'll each be stocked with a different brand, and they'll have spare, labeled, unopened reams inside or near them, so you can identify the brand.
Paper is composed of a bunch of fibers - on a microscopic level, these look vaguely like a bunch of chunks of hay or grass mashed together. Or perhaps like a rough, hilly topography. When you draw with a pencil, the lines you make generally only show up on the peaks of those "hills", on the fibers that stick up higher than their peers. This means that there are little gaps in your lines. On really bad paper, these can become quite apparent, and can be so bad that, like grooves in a record guiding a needle, they cause your pencil to move in directions you don't want it to. That can foul up finer details in a drawing. If the paper is causing these problems, it's not good enough. I suggest buying the best ream (500 sheet pack) of laser printer paper you can find in your local department store. This should cost you no more than about $6-10.
Professional artists sometimes buy quality paper called "bristol board" or "vellum" for their illustrations (such as those mentioned at the start of this tutorial). It's only worth the money when you get really, really good at illustration; I myself am not good enough to capitalize on the difference, so I don't buy it.
Total cost for a ream of good paper: about $10
Your best friend.
A kneaded eraser is a must. This is your primary tool. These are versatile things, and you may find them very strange if you're unfamiliar with them. They can be deformed like putty, often useful for making a point with which to erase fine details (a task they excel at), and also do not leave "shavings". They pull the graphite into their own fibrous mass, removing it from the paper as you rub the eraser against it. Because of this, they will over extended use become filled with graphite, and thus useless for pulling more out - it's at that point that you throw them away.
Other erasers can also be very useful, including the white ones attached to most mechanical pencils. I use an "art gum" eraser for any "nuke and pave" tasks, wherein I want to erase a section of a drawing back to as close to pure white paper as I can get. It leaves a lot of shavings, and is a unwieldy tool for fine details, but does that single job very nicely. White vinyl erasers, and many other various brands/varieties can be used for similar purposes. Kneaded erasers, especially after some use, don't work especially well for nuking something down to white, and thus a few other erasers to complement them works well. The non-pink eraser at the end of my pencil works rather similarly.
You should avoid the ubiquitous "pink erasers" often seen in school classrooms, and also avoid anything harsh enough to tear up the page (like certain erasers I've seen which are designed to erase pen markings). The classic pink erasers don't do so well at lifting the graphite off the page, and sometimes have a detrimental effect of grinding it under the fibers, at which point you can no longer erase it without removing a layer of the paper, which is often unfeasible or destructive. They also have a tendency to dry out with age, at which point they're largely unable to erase, and will cause damage to your drawing if you try to use them.
Total cost for a batch of three erasers: about $10
If you want to color drawings on a computer, which I don't suggest trying until you can do really good pencil lineart, then you will need a few things. Any computer of about 500mhz, with a copy of the Gimp or Photoshop will do well. In 2006, these are trivial to acquire - people throw these kinds of machines away, and you might be able to get one for free, if you don't have one already. That underscores the real point, here, which is that the machine you're reading this on is more than likely grossly overqualified for doing this kind of work. You'll also need a scanner, which will set you back about $100 for a decent one, and I very strongly suggest getting a graphics tablet, which costs about $100 as well. Wacom sells the very nice Graphire tablets, of which the 6x8 size is perfectly adequate for anyone's needs (Craig Mullins has done his work with tools worse than what I listed above, as have many other artists).
For drawing in a cell-shaded style like this, I do not suggest coloring with anything other than a computer. People used to use things called screentones to color cell-shading art in the past, but computers are better than them in practically any way, especially efficiency of work, and being able to correct mistakes..
Total cost for a great scanner and tablet: less than $200
Total cost for a suitable computer: free, you probably have one.
Total cost for drawing software: free