Something that I forgot to mention: once you put a line down using a Calligraphic Brush you can adjust the overall thickness of the line by changing the stroke weight (look for the Stroke panel on the right).

If you want lines that are thinner than those in the demonstration, you might want to consider making your canvas much larger. A 3 pt diameter brush, like what I did in the video, is about as small as I'd like to go because the diameter settings for both diameter and variation are in integers. So if that 3 pt brush size is twice as thick as you want, make your canvas twice as big—or bigger. Why not. It's vector anyways.