Is this correct?
She is my mother. She always prepares breakfast and dinner for me. Dealing with the big things in my life every day, and paying a lot for me. She took care of me when I was sick. She cared about me when I was in a bad mood. She taught me when I was doing something wrong. She is the person who loves me the most in the world. Although I sometimes quarrel with her, I really love her. I hope one day I can be mature and make her proud. I also hope she can be happy every day and stay healthy.

Question

Answer

Very nice thoughts! You did a good job!
I would change only these two sentences.

1) Dealing with the big things in my life every day, and paying a lot for me. 
This sentence needs a subject. (SHE or MOTHER)
You could say, SHE deals with the big things in my life and pays a lot for me".
OR
"Dealing with the big things in my life, SHE pays a lot for me".
OR
"Dealing with the big things in my life, MY MOTHER pays a lot for me".

2) She taught me when I was doing something wrong. 
This sentence could be clearer in its TENSE. 
You could say, " She CORRECTED me when I DID something wrong". (past to match the other sentences)
OR "She TAUGHT me when I DID something wrong".

Hope this helps