This was a fun project to do! I had been shifting over to cartoons even before it was assigned, so I was thrilled.
As the directions would suggest, I first drew the cartoon on paper with pencil, outlined it in Sharpie, and scanned it onto the computer. From there I used Adobe Illustrator's pen tool to outline the lines bit by bit, and colored the white spaces between them through the lines' fill. In a few places I had to fill in the color by hand, as the lines' fill couldn't cover all of it. After that, I switched over to Photoshop to fill the rest of the white spaces I couldn't in Illustrator, and that was it for the flats. The last step was to shade everything with the Dodge and Burn tools.
I've found that the pen tool is used to make crisp and clear lines. It's more advanced than using the plain paintbrush, and handles curves easier than the Line tool. Although, it can't do quite everything-- for circles I had to use the Circle tool, and for filling in white space I had to break out the Paintbrush. Thus, I suggest using the Shapes tool with the Pen, and also the Paintbrush if there's any white space one needs to fill.
I did the lion's share of coloring in Illustrator, so I can't say much about Photoshop's paintbrush based on this project alone. I have, however, used the Photoshop paintbrush before in personal projects, so I am qualified to say that it's used pretty literally behind the lines layer to color things in. The Dodge and Burn tools are also used as a brush, but they darken or brighten the image's colors to create highlights and shadows. I wouldn't call them a paintbrush as such.