"I want to be doctor"
"I want to become a doctor"
Some might say that become is a more formal way of saying the same thing. But there is a slight difference. If you were auditioning for an acting role in a hospital drama, you might express your desire to "be" a doctor. It would sound awkward, although not necessarily wrong, to audition to "become" a doctor in a drama. "Become" suggests an application of something to arrive at a condition.
If you jumped in freezing water you would become cold. If you went to medical school you might become a doctor.
After the water has acted upon you, you would be cold.
After accepting your medical degree, you would then be a doctor.
It's a little tricky. You could say, "After the water has acted upon you, you would become cold", but it's better here to use "be". Consider...
"Hey, you there in the water! Are you cold?"
"No, but I'm becoming cold."
"So what if you stay in the water for a while longer?"
"Then I will BE cold!"
There's more to it, of course, than this - but it's a start.
Student or Learner