WOOCOMMRCE PRODUCT CATEGORIES

show product categories 

<?php
    echo $product->get_categories(
        ', ',
        '<span class="posted_in">' . _n( 'Category:', 'Categories:',
        sizeof( get_the_terms( $post->ID, 'product_cat' ) ),
        'woocommerce' ) . ' ',
        '.</span>'
    );
?>

loop products in category 

<ul class="products">
    <?php
        $args = array( 'post_type' => 'product', 'posts_per_page' => 1, 'product_cat' => 'shoes', 'orderby' => 'rand' );
        $loop = new WP_Query( $args );
        while ( $loop->have_posts() ) : $loop->the_post(); global $product; ?>

            <h2>Shoes</h2>

                <li class="product">    

                    <a href="<?php echo get_permalink( $loop->post->ID ) ?>" title="<?php echo esc_attr($loop->post->post_title ? $loop->post->post_title : $loop->post->ID); ?>">

                        <?php woocommerce_show_product_sale_flash( $post, $product ); ?>

                        <?php if (has_post_thumbnail( $loop->post->ID )) echo get_the_post_thumbnail($loop->post->ID, 'shop_catalog'); else echo '<img src="'.woocommerce_placeholder_img_src().'" alt="Placeholder" width="300px" height="300px" />'; ?>

                        <h3><?php the_title(); ?></h3>

                        <span class="price"><?php echo $product->get_price_html(); ?></span>                    

                    </a>

                    <?php woocommerce_template_loop_add_to_cart( $loop->post, $product ); ?>

                </li>

    <?php endwhile; ?>
    <?php wp_reset_query(); ?>
</ul><!--/.products--> 

-------------------------------------------------- 

<?php

$args = array(

    'number'     => $number,

    'orderby'    => $orderby,

    'order'      => $order,

    'hide_empty' => $hide_empty,

    'include'    => $ids,

     'parent'    => 0

);

$product_categories = get_terms( 'product_cat', $args );

foreach( $product_categories as $cat ) {

echo '<li><a href="'. get_site_url().'/product-category/'. $cat->slug .'">'. $cat->name . '</a></li>';

}

?>

 --------------------------------------------------

get categories for a specific product given the id

global $post;
$terms = get_the_terms( $post->ID, 'product_cat' );
foreach ($terms as $term) {
    $product_cat_id = $term->term_id;
    break;
}