I always thought malen is to draw and färben is to paint but then apparently there's this zeichnen too. Anyone mind sharing their thoughts?
zeichnen = to draw
You use a pen or pencil and draw lines on paper or other flat and smooth surfaces. If you want to fill areas, you have to hatch those areas (to hatch = schraffieren). Without hatching the result of »zeichnen« are only thin lines. This result is called »Zeichnung« (english: drawing). A typical Zeichnung is monochromatic, but if you use colored pens, you also can get colored drawings.
malen = to paint
You use a brush, tip it into a colored liquid and then you use the color-soaked brush to applicate the color onto a more or less stabile surface. (You can not paint a picture on the hair of a woman.) You fill areas when you are painting, those areas build a pattern that will be interpreted as a picture. If you want to get thin lines, you have to use a thin brush. The suface on which you applicate the color can be smooth or also rough (like stone, or a wall made of bricks). The result is called »Gemälde« (painting) or often just »Bild« (picture). A typical Gemälde is not monochrome, typically you will see lots of colors in a painting.
färben = to dye, to color
You do not applicate color onto a stabile surface, and normally you don't use a pen or brush for färben. You dye/color a fabric or hair. Typically you dunk the fabric into a colored liquid. If you dye hair, you applicate the color onto the hair, and you can do this with a brush, but since hair is not a flat surface, you will not get a colored pattern that will be interpreted as a picture.
The distinction is a technical one: "malen" is to paint with a brush and colour and "zeichnen" to draw with a pen or pencil. However, in everyday usage, many people (and especially children) mix up the concepts and say "malen" when using a pencil. The reverse is less common, though.
While "malen" can be both creative as an art form or just mean applying colour to something, "färben" does not carry the additional meaning of an art technique.
I often see "zeichnen" used to express a way to draw clear lines. Often it's either meant for technical or skillful drawing.
While "malen" originally describes the art of painting, it's very common to use it for unskillful or untechnically drawing, but not in a disrespectful way ( that would be "kritzeln" or "gekritzel" ).
Examples: "Meine Schwester kann gut zeichnen." - My sister is good at drawing
"In der Grundschule haben wir oft mit Buntstiften gemalt." - In elementary school, we often drew pictures with (color-)pencils
"Kannst du das mal aufmalen? Ich kann mir das nicht vorstellen." - Can you draw me a picture? I can't visualize it.
"Ich habe eine Zeichnung von dem Gebäude angefertigt." - I created a drawing of the Building
@Hubert Schölnast: Very nice explanation!
I might add just one thing: Zeichnen can be a very creative, and original art, as well as a very exact craft (down to the millimeter, as needed in technical drawings / technischen Zeichnungen). The result of malen can be very detailed but will not be exact.