how to display all categories in homepage

why don’t you buy the theme?

generally:
use get_categories() and then loop through them, with a foreach loop; make ‘mini-loops’ for each category, using possibly get_posts() with the 'catgegory__in' and 'numberposts' parameter; styling the first post in these ‘mini-loops’ different as the three following posts; html structure with divs with fixed widths and float:left;

possibly similar question:
http://wordpress.org/support/topic/latest-5-posts-from-each-category?replies=20
http://wordpress.org/support/topic/how-to-display-5-latest-post-title-on-the-main-content?replies=4

This is interesting, would like to know as well. Although, I don’t have PHP skills.