Today is Korea public holiday. and I don't go to academy on English.

These day I'm studing Toeic and searching for my comfortable field work.

I always have concern about my future. and i should study English to my priority.

concern about my future 

I am concerned about my future.

Today is Korea public holiday. and I don't go to acadamy on English.
Today was a Korean public holiday, and I didn't have to go to the English Acedemy.

Theseday I'm studing Toeic and searching for my interested field

These days i've been studying for Toeic and searching for my interested field.

I always have concern about my future. and should study English and major

I have always been concerned about my future. Should I study english as my major?

concern about my future

Today is Korea('s) public holiday.

or (Today is a public holiday in Korea) 

and I don't go to acadamy on English (English Academy)

Theseday (These days)  I'm studing (studying) Toeic (Toefl?) and searching for my interested field 

I always have (been) concerned (or I have always been concerned) about my future.

and should study English and major 

I am confused what you are trying to say in this sentences. 

are you wondering if you should study English as a major?

I want for you to find my mistake (I tried *wink) 

Good luck, I know the confusion and frustration that can come with choosing the right major. What I can suggest you is to be honest to yourself. Try to find out what is that course that you can study for hours without getting tired from it? 

(I agree with most of Mikaela's corrections, but I found a few more)

Concerned about my future

Today is Korea's public holiday and I don't want to go to acadamy to study English.

These days I'm studying for the Toeic test and searching for something to pursue. 

I am always have concerned about my future, so, I should study English and my major.  (you can also say: I have always been concerned about my future.)

I want for you to find my mistakes.