After adding the properties (indicated by comment: /* Required to expose to WP-JSON API */ ) to your register_post_type() function, you should be able to get your CPT data as JSON by visiting example.com/wp-json/wp/v2/cpt-endpoint.

Expose Custom Post Type (And Taxonomy) as Endpoint to WP JSON API

//Register the post type
register_post_type('podcasts', array(
  'label' => 'Podcasts',
  'description' => 'A library of podcasts.',
  'public' => true,
  'show_in_rest' => true, /* Required to expose to WP-JSON API */
  'rest_base' => 'podcasts', /* Required to expose to WP-JSON API */
  'rest_controller_class' => 'WP_REST_Posts_Controller', /* Required to expose to WP-JSON API */
  'show_ui' => true,
  'show_in_menu' => true,
  'capability_type' => 'post',
  'map_meta_cap' => true,
  'hierarchical' => true,
  'rewrite' => array('slug' => 'podcasts', 'with_front' => true),
  'query_var' => true,
  'menu_icon' => 'dashicons-controls-volumeon',
  'supports' => array('title','editor','excerpt','thumbnail','post-formats'),
  'labels' => array (
    'name' => 'Podcasts',
    'singular_name' => 'podcast',
    'menu_name' => 'Podcasts',
    'add_new' => 'Add Podcast',
    'add_new_item' => 'Add New podcast',
    'edit' => 'Edit',
    'edit_item' => 'Edit Podcast',
    'new_item' => 'New Podcast',
    'view' => 'View podcast',
    'view_item' => 'View Podcast',
    'search_items' => 'Search Podcasts',
    'not_found' => 'No Podcasts Found',
    'not_found_in_trash' => 'No Podcasts Found in Trash',
    'parent' => 'Parent podcast',
    )
  )
);

// Register the taxonomy
register_taxonomy( 'podcast_categories', 
array(0 => 'podcasts'),
array(
  'show_in_rest' => true, /* Required to expose to WP-JSON API */
  'rest_base'=> 'podcast-categories', /* Required to expose to WP-JSON API */
  'rest_controller_class' => 'WP_REST_Terms_Controller', /* Required to expose to WP-JSON API */
  'hierarchical' => true,
    'label' => 'Podcast Categories',
    'show_ui' => true,
    'query_var' => true,
    'show_admin_column' => false,
    'labels' => array (
      'search_items' => 'Podcast Category',
      'popular_items' => '',
      'all_items' => 'View All',
      'parent_item' => 'Choose Parent',
      'parent_item_colon' => '',
      'edit_item' => 'Edit Item',
      'update_item' => 'Update Item',
      'add_new_item' => 'Add New',
      'new_item_name' => 'New Item Name',
      'separate_items_with_commas' => '',
      'add_or_remove_items' => 'Add or Remove',
      'choose_from_most_used' => '',
      )
    )
  );