A new school year is coming... there are just a couple of days of vacation left (or "there are just a couple of vacation days left" -- to avoid using "of" multiple times) and on September (the) first I will have to go back to school. I had two months of vacation but it passed (or "went by") fast. This summer was not like past Summers. The weather has changed a lot and we did not have a lot of sunny days. It was often raining and it influenced (or "affected") my mood. Otherwise, I have good memories from this vacation. My cousins visited me and we had a great time together. This Summer I also learned Hiragana and now I know it. I am happy because I will be back to school and I will see my friends. I think it is time to learn something new, it was a lot of time to rest. The fact that we will have a lot of tests soon does not make me happy but that is normal in school life.
This year I will not have chemistry and Latin and it is wonderful for me, because I've always disliked these subjects. Mathematics and physics will still be subjects which (or "that" or "") I have to learn, and I hate that, but there are many other subjects which I am glad to learn like history, Serbian language and literature, English, Italian...
I expect this school year will be as successful as the previous. However I will do my best and I expect good results. :))

Very good job! I just added a few articles, changed a verb tense or two, and capitalized names of seasons and languages (and nationalities). Please note that in spoken English, we usually use contractions.. not using them makes your writing look a bit more formal. In general, your language is more formal than that used by your U.S. counterparts, which some people would consider commendable. Keep up the good work, Kiki!