Do they mean the same thing? Are they correct?
..working while going to school..
..working during school...

I would like to say that I'm a student and I have a part-time job too.

Question

Do they mean the same thing? Are they correct?
..working while going to school..
..working during school...

I would like to say that I'm a student and I have a part-time job too.

Answer

To achieve what you are trying to say, "working while going to school" is the correct answer. 
"Working during school" sounds like you are in class and working online on a laptop. Or maybe working at a job during school hours.

Answer

In addition to your #1: "I'm working my way through school" is the most usual way to say it.

Or, "I go to school and work"

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