Pagination – not progressing

You should use WP_Query here, also, for pagination, you need to use the paged parameter, because WordPress need it to calculate the offset.

I guess you could replace both queries with WP_Query, and to avoid showing the first post twice, save the ID in a variable and pass it to the post__not_in parameter in the second query.

    $paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
    $args = array(
      'posts_per_page' => 10,
      'paged' => $paged,
      'post__not_in' => $first_post

    );

<?php query_posts($args); while(have_posts()) : the_post(); ?>       
    <?php echo get_avatar( get_the_author_meta( 'ID' ), 125 ); ?>
    <?php the_title(); ?>
    <?php the_time('F jS, Y'); ?>
    <?php the_author(); ?>
    <?php the_excerpt(); ?>
<?php endwhile; wp_reset_query(); ?>

And $first_post is an array with the id of the first post. You can set it like this:

$first_post = array(get_the_ID());

This code should be in the first loop.

This is using your code, which isn’t recommended, but it should work anyway.