Okay, assuming you only want to restrict the categories for the front page, but still list all categories everywhere else, just add the according template file to your theme:
front-page.php
<?php get_header(); ?>
<?php $my_query = new WP_Query('cat=79,120'); ?>
<?php if ($my_query->have_posts()) : ?>
<div id="post-area">
<?php while ($my_query->have_posts()) : $my_query->the_post(); ?>
<div id="post-<?php the_ID(); ?>" <?php post_class(); ?>>
<?php if ( has_post_thumbnail() ) { ?>
<div class="gridly-image"><a href="https://wordpress.stackexchange.com/questions/93684/<?php the_permalink() ?>"><?php the_post_thumbnail( 'summary-image' ); ?></a></div>
<div class="gridly-category"><p><?php the_category(', ') ?></p></div>
<?php } ?>
<div class="gridly-copy"><h2><a href="https://wordpress.stackexchange.com/questions/93684/<?php the_permalink() ?>"><?php the_title(); ?></a></h2>
<p class="gridly-date"><?php the_time(get_option('date_format')); ?> </p>
<?php the_excerpt(); ?>
<p class="gridly-link"><a href="https://wordpress.stackexchange.com/questions/93684/<?php the_permalink() ?>"></a></p>
</div>
</div>
<?php endwhile; ?>
</div>
<?php next_posts_link('<p class="view-older">View Older Entries</p>'); ?>
<?php get_footer(); ?>
Now you may completely change whatever you like for the front page, without affecting other archives.
However, if really all you need to change, is listing either just the two or all categories, you could also stick with your index.php, and check the HTTP request and act upon that:
if ("https://wordpress.stackexchange.com/" === $_SERVER['REQUEST_URI']) {
$my_query = new WP_Query('cat=79,120');
if ($my_query->have_posts()) :
echo '<div id="post-area">';
while ($my_query->have_posts()) : $my_query->the_post();
} else {
if (have_posts()) :
echo '<div id="post-area">';
while (have_posts()) : the_post();
}