In addition to the corrections given by Rainy please note:

"There are three people in my family: my father, my mother and me."

To use "were" implies that there are no longer 3 people in my family (maybe somebody has left or died). 

Take care,

Josephine

First notebook entry: Introducing myself

My name is Wang Siqi, I am a Grade Seven student. I like swimming , drawing and English! There were three members in my family including my father, mother and me. At weekend, I am used to visiting my grandmother, sometimes I will go to hiking with my parents.
That's me, a lovely girl.
Why we should learn English ?
Sometimes, we ask ourselves: “Why do we (have to) learn English?” then the answer is, simply, to succeed in the exams. Everybody knows that English is a good medium of communication, within around the world, so we must learn it.

Why do we have to learn English? I putted "have to" in blue because it is not necessary but personally I would use it.

Also phrase "within the world" is not the way english people would say it. They would prefer to say "around the world". It would also be possible to say "on the world" but that way it don't follow context as much.