Learning how to use measure / count words properly
Here has been my (I think successful) strategy:


Use measure words for mass nouns accurately. This is exactly like English:


  一杯水 = a cup of water
  
  一瓶可乐 = a bottle of cola
  
  一斤青菜 = a half-kilo of cabbage


You really have to use these in all languages, because you can refer to really different quantities depending on the measure word (a tank of water vs a drop of water).
Learn the measure words that are based on shape. This is really different from English, but in spite of being challenging, they're rather important. Chinese people use them relatively consistently and might have trouble understanding you if you don't use them properly (or at least will find it annoying if you don't). Also, they're not that hard to learn, and you quickly develop an intuition for what kind of objects take what kind of measure words:


  条 -- long, thin, windy things
  
  
    一条路 = a road
    
    一条裤子 = a pair of pants
    
    一条蛇 = a snake
  
  
  支 and 根 -- stick-like things, generally stubbier than 条 things, and not flexible
  
  
    一支笔 = a pen
    
    一根香烟 = a cigarette
    
    一根灯管 = a fluorescent tube (lightbulb)
  
  
  张 -- flat, broad, thin things
  
  
    一张桌子 = a table
    
    一张纸 = a sheet of paper
    
    一张光盘 = a DVD (or CD)
  
  
  颗 -- pebbly, kernely, small-and-round things
  
  
    一颗牙齿 = a tooth
    
    一颗星 = a star
    
    一颗心 = a heart
  

Use 个 for everything else until you've been corrected a few times. There are a lot of weird measure words that are only ever used with one or two nouns. Even a lot of Chinese people use 个 in those circumstances. There's no point learning these if you don't have to, so let experience be your guide for which other measure words to learn.


EDIT: Just to be clear, I'm not advocating only learning 4 measure words. I just think that the less common ones should be learned through real life experience rather than memorizing word lists. Also, I should point out that sometimes different measure words can be used on the same noun, but they mean different things. When confusion arises, learn the distinction:


  一门课
  
  
    A subject or course (as in "I have 3 classes this semester")
  
  
  一次课 
  
  
    A meeting-time or occurrence of class (as in "When's the next class? Next Monday.")
  
  
  一节课/一堂课 
  
  
    Classes often have a 10 minute break in the middle. The first and second halves can be referred to this way. ("When does the next section start? In 15 minutes.") I think that 节 and 堂 sometimes overlap with the meaning of 次 when they're not being used contrastively. Correct me if I'm wrong.