I am eager to improve my speaking skills in Korean and English speaking skills.

It's correct originally, but with this word order it will sound more smooth (in my opinion

And The more I watch Eric Nam's interview videos, the more I want itmotivated I am.

"I want it" is okay, but I reworded to make it more clear.

That is whaty I want to take a leaf out of his book.

So I decided to watch Ted and write summary and my thoughts at least 1 in a week.

If Ted is referring to "TED Talks" and not a person, it should be capitalized (I'm not sure what or who Ted is 🤔

That must be worth to do itdoing.

I don't know what it is exactly is called.

And I have to take a test,the TOEFL test to improve my English skills and to apply to universities abroad for graduated school.

I am not sure what I am going to do but I am considering a lot of selecoptions.