Can I Make specific posts not show on the home page?

see:
http://codex.wordpress.org/Function_Reference/query_posts
http://codex.wordpress.org/Function_Reference/WP_Query

or try it with this in your theme’s functions if your home page is not a static front page:

function my_post_queries( $query ) {
  // not an admin page and is the main query
  if (!is_admin() && $query->is_main_query()){
    if(is_home()){
      $query->set('category__not_in', array(23,66));
    }
  }
}
add_action( 'pre_get_posts', 'my_post_queries' );

Change 23,66 to the category IDs you want to exclude on the home page