Building off of Ahmed's corrections...

First notebook entry: Introducing myself

I am learning English because I want to live and study abroad. 
My native language is Persian and I grow up in Iran. 

You would want to say "I grew up in Iran." Grow is the present and grew is the past, so if you were talking about your childhood, grew is the appropriate form.

If you grew up in Iran and are still living there and that's what you're trying to say, go with "I live in Iran" or "I have lived in Iran my whole life" to really emphasize the point.

I significantly enjoy studying and learning English.

This is correct, but it's an odd word to use in this context. Try "I really enjoy..." or "I enjoy studying and learning English a lot."

First notebook entry: Introducing myself

I am learning English because I want to live and study abroad.
My native language is Persian and I grow up in Iran.

I significantly enjoy studying and learning English.