previewing my posts on static page?

You will need to set a Static Page as your front page, create and assign a PHP Template for that page, and stack multiple Queries in the template to search out your required categories and display them. Here is one, you might have many:

// get the last post (any date) from 'Featured' category
$featured = new WP_Query( 'category_name=Featured&posts_per_page=1' );
if ( $featured->have_posts() )
{
    while ( $featured->have_posts() )
    {
        $featured->the_post(); // fill $post with data
        ?>
            <div id="post-<?php the_ID(); ?>" <?php post_class(); ?>>
                <!-- output markup and display the post(s) -->
            </div>
        <?php 
    }
}
rewind_posts();

// start next query

That should give you an idea, at least, there is a fair bit to it. Good luck.