Hi Ravila - I will correct this sentence by sentence. :) 

Y = you wrote

C = corrected version

Y: I live and study in Moscow and my home in Petrovsk. 

C: I live and study in Moscow, but I am originally from Petrovsk. 

Y: Four days ago I passed last exams and come home. 

C: Four days ago, I passed my final exams and came home. 

(In the USA, we call "last exams" "final exams". You can also say "finals" for short and people will know what you are talking about. ;) Also you want to use the conjugation "came"... because you are talking about the past. "Come" would be present tense.) 

Y: I want to go to traines of athletics and table tennis. 

C: I want to speak with the athletic trainers and the table tennis coaches. 

(Note: In America, you often see "ping pong" written instead of "table tennis". They mean the same thing.) 

Y: When I was at school I trained. 

C: I trained when I was at school. 

Y: I love my trainers, because they a lot teached me. 

C: I love my trainers because they teached me a lot. 

(A matter of word order)

Y: Also I would like to learn English: I will watch films in english and will write notebook here. 

C: Also, I would like to learn English. I will watch films in English and I will write notebooks here. 

I hope this correction helps you ~ you are doing great! :)

My plans for holidays

I have been living and studying in Moscow and but my home is in Petrovsk. Four days ago I passed my last exams and I have come home. I want to go to traines coaches of athletics and table tennis. When I was at sсhool I trained. I love my trainers[coachers], because they taught me a lot teached me. Also I would like to learn English: I will watch films in English and will write in the notebook here.

Firstly Ravilya, thank you for using "Holiday". (Instead of the American word 'vacation'.)

A person who trains athletes is usually known as an "Athletics Coach", often shortened to "Coach" and this term is applied to most sports.

What were the subjects of your exams?

It seems you already have some knowledge of English. I'm sure being here on italki will help.

Den

My plans for holidays

I live and study in Moscow but I am originally from Petrovsk. Four days ago I passed my final exams and came home.  I want to speak with the athletic trainers and the table tennis coaches. I trained when I was at school. I love my trainers because they taught me a lot. Also I would like to learn English: I will watch films in English and will write notebooks here.