Currently, I longed for athe opening day of the movie "Violet Evergarden".

As I mentioned before, tThe story is about a fourteen-years-old girl named Violet, who loses/has lost both her arms and her emotions by fighting as a soldier, but learns to express her true feelings little by little and make people around her happy.

I'd like to use these words orand phrases, (which I'm not likely to use, often) as much as possible in order to expand my vocabulary.

CurrRecently, I have been longeding for athe opening day of the movie "Violet Evergarden"."

"Currently" means now, immediate, and only now. It is also generally used with the present tense.
"Recently" means both relatively soon in the past and the current time.

We use "have been" here to express present perfect progressive tense. It's used to describe things that have happened in the past, are still happening now, and may continue into the future--exactly how you are feeling about this movie!

We change "a" for "the" because you are talking about a specific movie on a specific date. "The" denotes specificity. If you were talking about any old movie, you could have used "a." But since you are only talking about one movie, Violet Evergarden, you use "the."

S

(Would it be better to say "I can't wait for athe day..." be better? "?)

Again, talking about the specific day--you use "the" here.

The rest of the sentence is technically grammatically correct, but a little hard to read. Generally, you try not to insert things into the middle of a phrase if you can help it. So here, we rearrange the words to form a more coherent sentence.

As I mentioned before, the story is about a fourteen-years-old girl named Violet, who lost both her arms, ands well as her emotions by, from her time fighting as a soldier,. Throughout the story, she learns to express her true feelings little by little, and maklong with making the people around her happy.

"Fourteen-year-old" : In English, this can be a little tricky. But a rule of thumb is: If you can replace the age with just a number, you use "years" instead of "year". If not, you use "year.
For example: I am fifteen years old. -> I am fifteen.
The above sentence makes sense both ways, so you have the plural, or "years".
Example 2: The fifteen-year-old girl went to high school. -> The fifteen girl went to high school.
Here, the second sentence doesn't make sense, so there is no plural; you use "year".

This sentence here is also a "run-on sentence," or a sentence that has gone on too long without proper punctuation or grammatical accuracy. In English, you want to split the sentences up a bit; leaving them too long makes them hard to read. We should also split run-on sentences by topic. So we split it right after we describe Violet, but before we describe the story.

Speaking of describing Violet, she's a tricky one! Because she did lose both of her arms, not just one arm, you can write "both" and have it apply to not just the inclusion of her loss of emotions, but also the fact that all two of her arms were lost. Therefore, we want to make it clear that she lost all of the above. So we say, "She lost both arms, AS WELL AS..." instead of grouping them together.

I took've taken a day off in order to see it as soon as possible, and. I also want to get a special booklet for visitors as well.

Normal past tense here isn't correct. Instead, use the present perfect form (have taken), because you took the day off for an event that is happening in the future. Rather, the day that you took off is in the future, so you can't use simple past tense to express it.