I'm dying to read Korean books these days!

Good. Natural sounding expression that is used very commonly.

I'm a big fan of Korean novels. After I came to the USA, I've never read Korean books.

Two alternatives:

1. Since I've been in the USA, I haven't read any Korean books.

2. I have not read any Korean books since coming to the USA.

(For #2, you sould also say: "I have not read any Korean books after coming to the USA," but that is not as clear as to where you are now. If you say "Since coming," it implies you are still in the USA, and I think this is what you mean.

I was searching on the google, and I found 'Please, look after my mom.'

I was searching on Google and I found "Please, look after my mom." (Two comments: Since "google" is a proper noun, you do not need a definite article (the). 2) single quotes are not usually used in the outer quote, so I replaced that with double quotes.)

What a wonderful book!

Excellent expression. Good usage. :)

Even though I can understand whole ideas of that book, I really wanna read in Korean.

I'm not certain I understand this, but what I think you are saying is: Even though I can understand the overall idea expressed in the book, I really just want to read in Korean.

I miss Korean! lol.

As a general expression, "I miss Korean!" is fine. However, there is a chance for it to be taken as unclear if someone does not catch the full context. You can be more specific and say, "I miss being able to read Korean!" or possibly "I miss reading the Korean language!" or something to that effect. When speaking informally to a friend where your friend already knows you are talking about reading, then just saying "I miss Korean!" would be fine and would be understood.

I know this isn't formal writing, but just want to point out that lol is only used in chat, message board, and very informal settings like that. It is fine in this case if you intend it to be informal, but in formal writing, you would not use such an expression.

Very good.

I'm dying to read Korean books these days!

I'm a big fan of Korean novels. After I came to the USA, I never read Korean books.

It's a little more natural to say, "Since I came to the USA, I haven't read any Korean books.

I was searching on the google, and I found 'Please, look after my mom.' What a wonderful book! Even though I couldn't understand all the ideas of that book, I really wanted to read in Korean. I miss Korean! lol.

((I'm not sure if you meant you could or could not understand the ideas))

I've been dying to read Korean books these days!

I'm a big fan of Korean novels. After I came to the USA, I've yet to have a chance to read a Korean book. I was searching on Google, and I found 'Please, Look After My Mom.' What a wonderful book! Even though I can understand whole ideas in that book, I really want read in Korean. I miss Korean! lol.