My name is Max and I live in Moscow. I'm 20 years old. I have been studying English for 6 months and I hope to both improve my English skills and look for new friends around the world!

Excellent! I assumed that your tone is somewhat relaxed and informal, so I have corrected the entry that way. If you meant to be more formal, I would suggest something very similar to Jeff's correction: 

My name is Max and I live in Moscow. I am twenty years old and I have been studying English for six months. I hope to both improve my English skills and make new friends around the world!

First notebook

My name is Max and I live in Moscow. I'm am 20 twenty years old. and I have been studying English for 6 six months. and I hope to improve my English skill and look for new friends around the world!

Basically, your English is not bad.  Just some minor punctuation errors.  If you have only studied the language just for six months, I can imagine how good it will be in six years! :)  Welcome to Italki.