Create categories with their own teasers and sliders I'm new to Drupal. First look at my site's structure:


Homepage


slideshow carousel - displays only selected nodes from the whole website
menu - displays links from a menu created for homepage
teaser - currently displays "promote to front page" items
footer - common to all pages (may be menu or site map)

Category


slideshow carousel - displays selected nodes only from current category
menu - displays links from a menu or subcategories with their nodes
teaser - selected nodes only from current category
footer - common to all pages (may be menu or site map)

Article


no slideshow
no menu
path to article (category breadcumb)
article title and body
footer - common to all pages (may be menu or site map)



How to divide website into categories as mentioned above?

Which modules to use? By default Drupal has Taxonomy module. Is it enough?

The simpliest way is to create custom templates like html--front.tpl.php, html--node--17.tpl.php, html--taxonomy--term--4.tpl.php, etc. but it's not flexible way. If you want to add more categories, you have to prepare another .tpl.php template.

What's the best and simple way to achieve this functionality?

More explanation:

Slideshow carousel is currently Slideshow:View module that displays all nodes with custom boolean field "promote to slider". Teaser on homepage works but I have no idea how to display it on category pages and even how to implement categories in Drupal. Any idea?
I would recommend a content type for the slideshow and the article. For the categorization do you use taxonomy and Views for filtering and sorting. 
I can recommend https://www.drupal.org/node/394094