Diary passage

In the first case, I would probably write:

I was confused: What should I write?

However you do it, you certainly need some punctuation between "confused" and "what". Alternatively you could say:

I was confused about what to / what I should write.

In the second case, since you are actually saying it to yourself, I would use quotation marks:

"What else is important to me?", I often say to myself.

Alternatively, you could say:

I often ask myself what else is important to me. 

There are one or two other things that need fixing in your text. Since I'm here, I have underlined them:

When I promoted to 2^nd grade, my father brought [could be correct] me a barbie cover diary. I loved pink and barbie as well. Its pink and purple lining pages encouraged me to write some thing special on it. ... I go to school and study usually, but I can’t write school notes on it. I have a common life. ... 

Cute572

I want to know why we don't use question mark in such context you have explained. I have seen this type before but i don't know the exact reason for usage.

"I was confused about what to write" (for example) is not a question, it's a statement. It therefore does not need a question mark. A sentence that is a question usually starts with one of the words What, Who, Why, Where, When, How, Which, Whose, or with the verb and the subject round the other way (for example, Am I happy?, rather than I am happy). (This isn't intended to be a complete list of ways to ask a question.)

When I got in to (the) 2^nd grade, my father bought me a beautiful diary. I loved its pink colored pages and in some way it encouraged me to write something special on in it. ... I go [If you're talking about your present situation then "go" is fine. If you're describing what happened in the past then you should use "went", but you'll have to change some of the other verbs too] to school and study usually ["Usually" here is not wrong, but I suspect it's not quite what you are trying to say], but I can’t write school notes on in it. I have a good life. ... ["good life" is OK, but has a completely different meaning to what you had before. Rather than choosing words randomly, it might be better to decide which idea you want to express, and then try to find the correct words to express it.]

Cute572

After this sentence we would continue same paragraph or start  with the new line? As after dialogs new paragraph begin but since it is the same speaker (me) what will we do here?

There is no need to start a new line here (it's not a dialogue). If you think it's a good place to start a new paragraph then you can, but it's not mandatory.

thanks