When chosing issue type I have taks, bug and story on the same level. They appear on board in the same way, they can only have sub tasks attached to them. Shouldn't it be this way that story is something bigger containing group of tasks?
story - logging in
tasks - logging screen, facebook login, email login
and subtasks - atomic activities connected to tasks
How should I use stories properly to have everything tracked properly?
In the example you give, I would suggest tasks and sub tasks are merged as all sub tasks of the logging in story.
For bugs relating to the story an option is to create a new issue type called story bug which is esentially a sub task of the story too. So when you create a sub tasks on a story you would have the option to select it as a sub task or a story bug.
At a higher level than stories you have epics which are broken out into multiple stories.
But I have only 2 levels still. Under story I would have big sub tasks like "facebook login" and "providing facebook login button" on the same level. I would like to have them on 3 levels- logging in-> facebook login -> facebook login button. Something like story -> task -> subtask. Is this order possible in some way?
Yes, epics are collections of stories.
Epics are not available in GreenHopper 6 yet. In v5 Classic, this page should help:
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