Also get implies obtaining/receving something.

I would like to get some food.

I can get that DVD.

I will fail the course if I get a bad grade.

In the past we would use "got"

I got some food (yesterday).

I got that DVD.

I failed the course because I got a bad grade.

Become is "being" in reference to the future.

I will become a better student.

He wants to become a pilot.

Became is "being" in reference to the past (and implies continuation until now).

She became a teacher at the age of 40.

He became a doctor 5 years ago.

<em>Get</em> has the meaning of <em>become</em> when an adjective follows.

<em>get angry</em>

<em>get better</em>

<em>get old</em>

...and so on... 

Just a thought (and this might be just me or something) but 'become' is kind of formal for everyday speech.

For example the stuff on the left is what I would say instead of the stuff on the right:

I'm gonna be (going to be) a doctor = I will become a doctor

I'm getting bored = I'm becoming bored

I'm hungry = I have become hungry

We've gotten pretty close lately = we're becoming friends

I can't think of when I would say 'become' in a sentence, unless I was being deliberately dramatic just for laughs.

Thanks for your answer.