My tragedy!!!

Three weeks have already passed since I took my last piano class (or lesson). After that, my teacher was very busy and we could not meet each other because of his lacking of free time. 

Here, you have explained that he is busy in the first half of the sentence so to say "because of his lacking free time" is just repeating the same information, namely "He lacks free time" is another way of saying "he is busy". Instead, try to join related information, but not the same information in sentences.

ie. After that, my teacher was very busy and so (as a result) we could not meet each other.

In my last class, he gave me the first piece of Czerny 299 as my piano exercise homework. However, I could not even play the whole movement completely—It is really too hard for me. So I made the decisions on my own, to drop Czerny 299, and pick up Czerny 849 again, and played many a lot of pop music instead.

However, the tragedy happened again, after playing the "Croatian Rhapsody", my left wrist pained me (either "pained me severely" or "gave me a lot of pain") severely.  through my After carefully checking, I found that my capillary had broken breaking. It is really a huge tragedy for me!
I gave a call to my teacher and he promised to took shift (or postpone) my piano class on Friday this week.

Well done Erin, keep up the good work, and I am sorry to hear of your painful wrists.
A.