how to “not” display new posts with a specific category on the main page?

There is even an example in the WordPress codex here for this:
http://codex.wordpress.org/Plugin_API/Action_Reference/pre_get_posts

function exclude_category( $query ) {
    if ( $query->is_home() && $query->is_main_query() ) {
        $query->set( 'cat', '-3,-8' );
    }
}
add_action( 'pre_get_posts', 'exclude_category' );

Just modify the category IDs (replace -3,-8 by your category IDs preceded by the minus sign) and put it in your functions.php or your plugin code.

The WordPress codex states the following:

query_posts() is the easiest, but not preferred or most efficient, way to alter the default query. The preferred way is hooking into pre_get_posts and altering the main query.

This way you just alter the main query instead of throwing away the results of the first query and running a second.