I agree with Vincent, the most efficient way to learn the measure words is to be familiar with them by using them frequently.
I say that because the original meaning of those measure words has evolved thousands of years, and even the native speaker do not understand their initial meaning unless they have to read ancient Chinese.
The initial meaning of 件 is to separate things. After separation, you can distinguish one thing from another. Thus, 这件衣服 are different from 那件衣服. And documents can be separated by the papers that is written. Therefore, 这件公文 is different from 那件公文. Each 公文 carries different purpose when people write them. As a result, 件 is used to measure the purpose or intention, 这件事 is different from 那件事, from concrete document to abstract event/intention.
Similarly, 把 is to hold things by your hand, as you can see the hand(手) at the left hand side of the character with 巴/ba1/ at the right hand side. So 把 is to count things that can be and should be hold with your hand：一把伞 一把铅笔 一把刀.
支 is to describe things that is thin and long like a stick. 一支笔，一支蜡烛，一支箭。