While most projects will be executed on Strathmore Sketch series 400 paper, longer assignments should be executed on a heavier paper. This can allow for more durability and workability without destroying the surface, as well as the potential to hold more charcoal. 
As a general note, a smoother paper will have less textural effects to overcome when rendering smooth subjects, but the lines of veiling may be more visible and it has a limited ability to hold additional charcoal to do many veils in dark passages. A more textured paper will require more effort in creating smooth passages, but will easily hold enough charcoal to render very dark shapes. 
Hot press paper is smooth, cold press paper is rough. 
Your paper should be pH-balanced and acid-free (a less expensive non-pH balanced paper can deteriorate) and 100% cotton (a stronger fiber paper) if possible. 
Some papers are softer (generally those intended for wet mediums) and have the potential to pill when worn down (this can be an advantage or disadvantage), while some papers are harder or stiffer (making sharp marks easier). 
While each paper has advantages and disadvantages, it is encouraged that you try a variety throughout the year to become familiar with working properties of each and how to properly adjust working methods as needed.
Ultimately, the paper should be suitable for the desired effect.  For example, detailed work should employ smoother paper, while a deeper range of darks should utilize a more textured surface. Additionally, if you know the weave of your desired canvas for painting, it can be advantageous to select a paper with a similarly textured surface.

In this program, you will work primarily utilizing 3 types of charcoal: willow charcoal, charcoal pencils, and compressed charcoal. 

Willow is made from the twigs of a willow tree, burnt to a precise temperature to establish the desired hardness or softness. It can be quite black, but it lacks binder, and therefore erases with great ease. Willow works well to establish and break down scheme lines due to its erasing ability; however, knots may be present which can scratch the surface. Scrubbing the willow on a sheet of paper or sand paper can effectively remove the knot. A very delicate touch is required to create a fine line with willow, as a heavy hand can produce thick, wobbly lines. You may sharpen the tip to ease the process of making points on the drawing. It is not advised to utilize a sharpened willow to create a line, though, as its sharp point may score the surface, and will deplete into a dull point almost immediately. Vine charcoal shares similar properties with willow charcoal but is typically lighter in value.
 

Charcoal pencils are made of charcoal powder bound into shape by either wax or clay and enclosed in a wood casing. The pencil will hold its point considerably longer than the willow and is therefore ideal for creating small details and precision marks. The use of binder renders the pencil more permanent than the willow, and as such should be used when accuracy has been established in the drawing with willow, or the student can command a very light touch with the pencil itself. The pencil tends to have a range more limited than compressed charcoal, however, and a full value range may not be available. 
 

Pencils are rated with a number system that indicates the hardness or softness of the charcoal. With HB as a baseline hardness, increases in the B number (B standing for blackness, 2B to 9B), creates a softer, blacker pencil. Increases in the H number (H standing for hardness, 2H to 9H), creates a harder, lighter pencil. While harder pencils are lighter, keep in mind they have a greater potential to score the paper, producing a non-removable mark. The scale is typically as follows: 
 

(Lightest and hardest) 9H, 8H, 7H, 6H, 5H, 4H, 3H, 2H, H, F, HB, B, 2B, 3B, 4B, 5B, 6B, 7B, 8B, 9B (darkest and softest). 

The F is an arbitrary number to indicate the hardness between H and HB. 

Not all brands of charcoal pencils are created with equal blackness. While the pencil rating scale listed above controls the volume of charcoal distributed (allowing the potential for darker marks), every company utilizes their own formula for charcoal, some being darker than others. 
 

Compressed charcoal is more permanent than willow and can often achieve greater depths of darks. It does not flow quite as easily as willow; however, it can be sharpened to provide greater detail. Compressed charcoal is utilized to achieve the darkest ranges of the value scale, when pencils and willow cannot complete the task. 

The kneaded eraser and the white retractable/click eraser, while both serving the function of removing charcoal, are utilized for very specific tasks, and at different stages of the drawing.

 
The kneaded eraser is a broad erasing tool, in that it removes charcoal with little subtlety. Although it can be formed to make precise points, it nevertheless removes large quantities. It must be cleaned frequently, through kneading, to ensure it will lift charcoal. If an excess of charcoal has built up in an area of the eraser, it will function more to push charcoal into the paper and smear it around (similar to stumping, though with less control). It should primarily be used early in the drawing, when the goal is to correct dark shapes (or produce light shapes within a dark area), or remove lines and charcoal completely. 
 

Another important function of the kneaded eraser is to keep the drawing clean (to avoid excessive stray marks and smudge buildup), particularly in the early stages of creating the envelope. This is accomplished by forming the kneaded eraser into a ball, then rolling it across the paper over areas to be cleaned. Any charcoal that is loose will easily lift off. This is done in favor of scrubbing with the eraser, which has the risk of pushing the loose charcoal into the paper.

The white retractable/click eraser is a sensitive tool in comparison to the kneaded eraser. It is encouraged to think of this eraser as a direct counter to the charcoal pencils, as it serves the same function, though in the capacity of making lighter veils. By sharpening the tip of the white eraser (with a razor blade), one can produce a similar precision to the pencil, though it will always fall short of precision and subtle value adjustment due to its less firm nature. It is able to produce veils resulting in twice or three times the value jumps of a pencil. That is to say, a charcoal pencil can produce veils two or three times more subtle. Considering this, one should always opt to veil down slowly to the correct value with a pencil, instead of attempting to arrive at the correct value immediately, making the drawing too dark, and needing to veil up with the eraser. It is a strong possibility that any time a light veil is applied with an eraser, correction will be needed with the pencil. 

Please note that this eraser also requires frequent cleaning. This can be done by lightly rubbing excess charcoal onto an adjacent paper, exposing the white of the eraser. If charcoal has built up on the tip, it will likely have the same stumping effect as a dirty kneaded eraser.