I'm very new here, and I have already found some people, who write good answers. I want to follow them. Like to watch their work in free time.
Is there any option to follow a user?
Every user has an RSS feed of their activity, which you can find at the bottom of the Activity tab of their profile page:
There is no other way to be notified of their activity within the Stack Exchange network (e.g. via inbox notifications). You'll need an RSS reader to consume this feed.