Display 3 level taxonomies

Here is general guideline to get what you need.
Use get_term() to get 1st, 2nd and 3rd level terms.Documentation here and here.

1st Level: you can display this list of terms directly in php as follows

$taxonomy = "SOME_TAXONOMY";
$terms = get_terms([
     'taxonomy' => $taxonomy,
     'parent' => 0,
     ]);


if ( ! empty( $terms ) && ! is_wp_error( $terms ) ) { ?>
<ul>
    <?php 
    foreach ($terms as $term) : ?>
        <li>
            <a href="https://wordpress.stackexchange.com/questions/332234/<?php echo get_term_link($term->term_id); ?>"><?php echo $term->name; ?></a>
        </li>
    <?php endforeach; ?>
</ul>

2nd / 3rd Level Terms
After displaying 1st level terms, you should use ajax action to get child terms of clicked parent term. Pass id of parent term to ajax action function. Documentation for using Ajax with WordPress is here.

Code for ajax action should look like this.

function wp_ajax_child_terms_action(){

    $taxonomy = "SOME_TAXONOMY";
    $parent = $_GET['id_of_parent_from_ajax'];
    $terms = get_terms([
        'taxonomy' => $taxonomy,
        'parent' => $parent,
     ]);


    if ( ! empty( $terms ) && ! is_wp_error( $terms ) ) { ?>
       <ul>
           <?php foreach ($terms as $term) : ?>
            <li>
                 <a href="https://wordpress.stackexchange.com/questions/332234/<?php echo get_term_link($term->term_id); ?>"><?php echo $term->name; ?></a>
            </li>
            <?php endforeach; ?>
      </ul>

<?php }  ?>

I hope this may help you start.

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