Show posts by category on Homepage

general approach:

possibly in index.php (or home.php – http://codex.wordpress.org/Template_Hierarchy ):

– get a list of category IDs or names – either manually added into an array, or automatically with get_categories() – http://codex.wordpress.org/Function_Reference/get_categories ;

– run a foreach loop through these categories, using a WP_Query() http://codex.wordpress.org/Class_Reference/WP_Query for each cat, and a loop to show the posts, and the first post different; wrap each category loop into a div for formatting as columns;

– add the corresponding css to make it match the design of your existing theme.

related forum topic:
http://wordpress.org/support/topic/latest-5-posts-from-each-category?replies=20