Only display latest custom taxonomy post

You should save the ID of the first post in a variable (e.g $splash_post_id) and in the next loop, use "post__not_in" => $splash_post_id , in the $args array.

$args = array(
    'post_type' => 'post',
    'posts_per_page' => '1',
    'order_by' => 'date',
    'order' => 'DESC',
    'tax_query' => array(
        array(
            'taxonomy' => 'theme',
            'field' => 'slug',
            'terms' => array ('text-image', 'just-image', 'just-text')
        )
    )
);

$query = new WP_Query( $args );
if (have_posts()) :
    while( $query->have_posts() ) :
        $query->the_post();
        $splash_post_id = get_the_ID();
          .......
          .......
    endwhile; 
endif;

$args = array(
     'post__not_in' => array($splash_post_id),
    'post_type' => 'post',
    'posts_per_page' => '10',
    'order_by' => 'date',
    'order' => 'DESC',
);

$query = new WP_Query( $args );
if (have_posts()) :
    while( $query->have_posts() ) :
        $query->the_post();

          .......
          .......
    endwhile; 
endif;