Hi Łukasz! I am glad you've decided to try and learn / speak English - here are a few corrections with your paragraph.

1) Yesterday I spoke first time by Skype in English witch (with) teacher from Philippines.

2) I (it) was very exciting, because my speaking (spoken - this one is difficult but all to do with tenses!) English isn’t good.

3) During this (the - you can use 'this' but it makes more sense to put 'the') conversation I said (spoke - as you were speaking about them) about me and my expectations.

4) The talk (conversation is a better word to use) with (a) person who didn’t knew (know) any word(s) in Polish is for me (more suitable to use this at the end of the sentence) (an) interesting and important experience for me.

5) That allow (This allowed) me to believe, that speaking with persons (speaking with someone) from other countries (from another country) is possible also (also possible - switch the words around) for me. ;-)

The final correction should read like this:

Yesterday I spoke first time by Skype in English with teacher from Philippines. It was very exciting, because my spoken English isn’t good. During the conversation I spoke about me and my expectations. The conversation with a person who didn’t know any words in Polish is an interesting and important experience for me. This allowed me to believe, that speaking with someone from another country is also possible for me. ;-)

I hope this helps!