I've highlighted the correct things in green! Good work!

I'm studying english for a half year, and all that i've learnt before it was from series, movies and music.

I have been studying English for six months, and before that learned only from TV series (shows), movies and music. 

I don't have to much difficult in listening and reading it, but i still have a lot of problems to write and speak properly.

I don't have much difficulty listening to or reading it, but I still have a lot of problems writing and speaking it properly. 

I decided to study it for fun at first, but nowadays english is a necessity., and I felt that i needed to learn it a few months ago, when a guy

Try to prevent what we call "run on" sentences. Notice the way I broke the sentence into two. ^^

sat by my side on the bus, he was foreign, and a don't remember which country he was, but he wanted information and i couldn't help

^^ This is NOT incorrect, but more of a mouthful than necessary. 

sat beside me on the bus, he was a foreigner, and I don't remember which country he was from, but he wanted information and I couldn't help him.

** In America, we use foreign to describe things and foreigner to describe people. **

However, you might also find that rephrasing this sentence makes it easier to say and the use of the original, foreign, more fitting. For example:

"I began to feel more strongly that I needed to learn English a few months ago; when I was unable to help a foreign man who needed information sat beside me on the bus."

him. We tried to talk, but I was very shy. After that i realize how much i had to improve my english.

There are a few ways that you could say this better, depending on what you mean:                                                         

>>>  After that, I realized how much I needed to improve my English. 

>>> After that, I realized how important it was to improve my English.

>>> After that, I realized how much work I had to do to improve my English. 

First note

In general this is really well written, job well done.