If you're interested in the original history of each, there is a good deal of information on the differences of all three by definition and origin. ie: http://www.englishteastore.com/shortb...
If made correctly, all three taste different and should appeal to what you're looking to bake. More butter flavor, more sugar, more crisp. Although modern bakers can blur the lines for all three types, the history of each is very clear and very interesting.
As understand it:
Shortbread is essentially a biscuit dough with added sugar and very high amounts of butter which makes it crumbly.
Butter cookie dough very much like shortbread but has more sugar making it less crumbly.
Sugar cookies are butter cookies topped with a layer of sugar when the come out of the oven.