Use Yoast primary category functionality to order primary category before other categories on frontend

get-modified-term-list.php

<?php
function get_modified_term_list( $id = 0, $taxonomy, $before = '', $sep = '', $after = '', $exclude = array() ) {
  $terms = get_the_terms( $id, $taxonomy );

  if ( is_wp_error( $terms ) )
    return $terms;

  if ( empty( $terms ) )
    return false;

  foreach ( $terms as $term ) {

    if(!in_array($term->term_id,$exclude)) {
      $link = get_term_link( $term, $taxonomy );
      if ( is_wp_error( $link ) )
        return $link;
      $term_links[] = '<a href="' . $link . '" rel="tag">' . $term->name . '</a>';
    }
  }

  if( !isset( $term_links ) )
    return false;

  return $before . join( $sep, $term_links ) . $after;
}

primary-category.php

<?php
function get_primary_category( $post = 0, $taxonomy = 'category' ) {
  if ( ! $post ) {
    $post = get_the_ID();
  }
  // SHOW YOAST PRIMARY CATEGORY, OR FIRST CATEGORY
  $category = get_the_terms( $post, $taxonomy );
  $primary_category = array();
  // If post has a category assigned.

  if ($category){
    $category_display = '';
    $category_slug = '';
    $category_link = '';
    $category_id = '';

    if ( class_exists('WPSEO_Primary_Term') )
    {

      // Show the post's 'Primary' category, if this Yoast feature is available, & one is set
      $wpseo_primary_term = new WPSEO_Primary_Term( $taxonomy, $post );
      $wpseo_primary_term = $wpseo_primary_term->get_primary_term();
      $term = get_term( $wpseo_primary_term );
      if (is_wp_error($term)) {
        // Default to first category (not Yoast) if an error is returned
        $category_display = $category[0]->name;
        $category_slug = $category[0]->slug;
        $category_link = get_category_link( $category[0]->term_id );
        $category_id = $category[0]->term_id;

      } else {
        // Yoast Primary category
        $category_display = $term->name;
        $category_slug = $term->slug;
        $category_link = get_category_link( $term->term_id );
        $category_id = $term->term_id;
      }
    }
    else {
      // Default, display the first category in WP's list of assigned categories
      $category_display = $category[0]->name;
      $category_slug = $category[0]->slug;
      $category_link = get_category_link( $category[0]->term_id );
      $category_id = $term->term_id;
    }
    $primary_category['url'] = $category_link;
    $primary_category['slug'] = $category_slug;
    $primary_category['title'] = $category_display;
    $primary_category['id'] = $category_id;

  }
  return $primary_category;
}
?>

project-categories-shortcode.php

function project_categories() {
		global $post;

    // Get the primary category array
    // If using a custom post type, replace second argument with custom post type taxonomy
    $primary_category = get_primary_category($post->ID, 'category');

    // Get the categories of the post, but exclude the primary category
    $terms = get_modified_term_list( $post->ID, 'category', '', '  / ', '', array( $primary_category[id] ) );

		$result = "<p class='category-list'>";
    $result .= "<a href='" . $primary_category[url] . "'>" . $primary_category[title] . "</a>";

    // If there are more than one terms, add the slash character to separate them from the primary term
    if ( $terms != '' ) {
      $result .= "  / ";
      $result .= $terms;
    }

    $result .= "</p>";
		return $result;
}
add_shortcode('project-categories',	'project_categories');