How to pick “full/thumbnail” images in the loop?

Try just using the traditional $i iterator (untested) but I can’t see why this wouldn’t work..

<?php 
$i = 0;
while ( have_posts() ) : the_post();
    $i++;
    if ( $i == 1 ): ?>
    <!-- First Post -->
    <div id="post">
        <a class="post_image" href="https://wordpress.stackexchange.com/questions/165568/<?php the_permalink(); ?>" title="<?php the_title(); ?>">
            <?php if ( has_post_thumbnail() ): ?>
                <?php the_post_thumbnail('full'); ?>
            <?php else : ?>
                <img src="<?php echo get_template_directory_uri(); ?>/images/No-Thumbnail-Available.png" alt="<?php the_title(); ?>" />
            <?php endif; ?>
        </a>
        <h2 class="post_title"><a href="https://wordpress.stackexchange.com/questions/165568/<?php the_permalink(); ?>" rel="bookmark" title="<?php the_title(); ?>"><?php the_title(); ?></a></h2>
        <span class="post_desc"><?php the_excerpt(); ?></span>
        <div class="clear"></div>
    </div>
<?php else: // else $i == 1 ?>
    <!-- All Other Posts -->
    <div id="post">
        <a class="post_image" href="https://wordpress.stackexchange.com/questions/165568/<?php the_permalink(); ?>" title="<?php the_title(); ?>">
            <?php if ( has_post_thumbnail() ): ?>
                <?php the_post_thumbnail('thumbnail'); ?>
            <?php else : ?>
                <img src="<?php echo get_template_directory_uri(); ?>/images/No-Thumbnail-Available.png" alt="<?php the_title(); ?>" />
            <?php endif; ?>
        </a>
        <h2 class="post_title"><a href="https://wordpress.stackexchange.com/questions/165568/<?php the_permalink(); ?>" rel="bookmark" title="<?php the_title(); ?>"><?php the_title(); ?></a></h2>
        <span class="post_desc"><?php the_excerpt(); ?></span>
        <div class="clear"></div>
    </div>
<?php endif; // endif for $i == 1 ?>
<?php endwhile; ?>