so happy

Today I had my hair cut. I like this style very much.

And today I was had been paid by/for from my part-time job in this winter vacation.

I like my life,my parents and myself.

Notes: It's a tricky sentence with a couple options to sound more fluent...

"And today I was paid for my part-time job this winter vacation."

You can also write: "And today I was paid for my part time work this past winter vacation."

OR: "And today I got paid for my part time work from this winter vacation."

Any combination of those sounds a little better. Be careful about using And to start a sentence. In strict English classes and rules, I was taught, a long time ago, that this is not formally acceptable, but nowadays we see it often.

Communication in the Modern World

Some of my friends believe that the modern technology gives us tremendous advantages. They can't imagine their lives without computers, tablets, smartphones and especially the Internet. On the one hand all of this variety of high-tech innovations is very useful for us, but on the other hand we are lacking a sense of real communication with people. If you need to find any information or you have some question, all you need to do is just to make a request on the Internet. In a second you get what you wanted. It is very simple, but in this way case we have become started communicating with each other much less than it was in the past.

Thank you.

NOTES:

You don't have write the Internet but it is more common that way.  It works both ways :)
I would say "variety" is singular so "innovation" should be too.

The last sentence is a little tricky.  I think it sounds more fluent to say started rather than become. It also might sound more fluent to say, "It is very simple, but in this way we have started to communicate with..."

Overall, very good job!