At the beginning of this time ofWhen I first started learning English, I rejectedfused to learning from teachers because I didn't want to pay for the learning, and I believed that I could learn English myself.

Finally, I found an online school which offered a cheaper price for lessons that (I thought could be/seemed) affordable.

In the first one or two months, I could only partly understand my English teachers, and so, I struggled to speak in English.

It was very hard to not be able to express the feeling that you wanted to tell people your ideas, but you were not able to, so I often got depressed, frustrated and even upset.

Although now I can speak English, I speak slowly, and often times I still have difficulties expressing myself.

At the beginning of this time ofWhen I started learning English, I rejected learning from teachers because I didn't want to pay for the learning, and I believed that I could learn English myself.

One year lateAfter a year, I found out that it did not help my speaking.

Sounds more natural

I was very excited whento pursue this idea popped up in my mind.

I have changed the meaning slightly, but it sounds more natural to me. Popped in my mind is usually used for a more sudden thought.

I realized that I cwouldn't not be able to speak English untilwell unless I practiced English with an Englishwith a teacher every day, so I started to book online lessons.

In addition, tenses, articles and other parts of speeches are still the most difficult part for me.

At the beginning of this time ofWhen I first started learning English, I rejected learning from teachers because I didn't want to pay for the learning, and I believed that I could learn English myself.

What you wrote isn't wrong, but it sounds awkward.

In addition, tenses, articles and partfigures of speeches are still the most difficult part for me.

If you mean non-literal expressions, the term for that is figures of speech.

At the beginning of this time ofWhen I started learning English, I rejected learning from teachers because I didn't want to pay for the learningclasses, and I believed that I could learn English myself.

One year lateAfter a year, I found out that it did not help my speaking.

But I still didn't want to buy online lessons, and I thought that if I could write well, I would speak well.

Great! This reads quite naturally.

I was very excited when I had this idea popped up in my mind.

I practiced writing in English for another year, but I still couldn't open my mouth to speak English.

Also very natural, good job.

I realized that I couldn't speak English until I practiced English with an Englishwith a teacher every day, so I started to book online lessons.

Your way works but saying "English" so much sounds weird, so this is more natural.

I was very hesitant before I decided to book online lessonse, because it was a brand new thingvery new for me.

Not incorrect, but this sounds more natural.

There were many things that I needed to take into considerationhink about.

Not incorrect either, but this sounds more natural.

Finally, I found an online school which offered a cheaper price for lessons, that I thought cwould be more affordable.

Not incorrect, but this sounds more natural.

In the first one or twofew months, I could only partly understand my English teachers and I struggled to speak in English.

To sound more natural

It was very hard to express the feeling that you wanted to tell people yourmy ideas, but you wereI was not able to, so I often got depressed, frustrated and even upset.

To sound more natural

Since the end of June of last year, I have been learning English from online English teachers for more than oneover a year, and now I can speak English more easily than before.

To sound more natural

At the beginning of this time ofmy English learning English, I rejected learning from teachers because I didn't want to pay for the learning, and I believed that I could learn English myself.

In the first year, I learned English in the same way as I did when I was in middle school.

"as" doesn't seem grammatically wrong here, it just feels awkward

I practiced writing in English for another year, but I still couldn't open my mouth to speak Englishit.

It was very hard to express the feeling that you wanted to tell people your ideas, but you were not able to, so I often got depressed, frustrated and even upset.

"even" suggests that the word after it will be even more severe than the words before it. But "depressed" and "frustrated" are stronger emotions than upset, so "even" doesn't make much sense here.
Better: "depressed, frustrated and upset."
Best: "upset, frustrated, and even depressed" or "frustrated and depressed"

Since the end of June of last year, I have been learning English from online English teachers for more than one year, and now I can speak Englishit more easily than before.

In addition, tenses, articles and parts of speeches are still the most difficult part for me.