How do I register a custom post type and taxonomy in WordPress the right way?

Register content types in a plugin, not the theme. If the theme is switched, content types in functions.php vanish and your posts become inaccessible (they remain in the database).

<?php
/**
 * Plugin Name: Projects CPT
 */
function acme_register_types() {
    register_post_type( 'project', array(
        'labels'       => array( 'name' => 'Projects', 'singular_name' => 'Project' ),
        'public'       => true,
        'has_archive'  => true,
        'rewrite'      => array( 'slug' => 'projects' ),
        'show_in_rest' => true,            // block editor + REST API
        'menu_icon'    => 'dashicons-portfolio',
        'supports'     => array( 'title', 'editor', 'thumbnail', 'excerpt' ),
    ) );

    register_taxonomy( 'project_type', 'project', array(
        'label'        => 'Project types',
        'hierarchical' => true,
        'show_in_rest' => true,
        'rewrite'      => array( 'slug' => 'project-type' ),
    ) );
}
add_action( 'init', 'acme_register_types' );

register_activation_hook( __FILE__, function () {
    acme_register_types();   // register first (init has already run), then flush once
    flush_rewrite_rules();
} );

Key points

• Post type keys: lowercase, max 20 characters, no spaces.
• show_in_rest => true is required for the block editor and REST access.
• After changing rewrite or has_archive, flush once (Settings > Permalinks > Save, or wp rewrite flush). Never flush on each request.
• Use a prefix or specific name to avoid collisions (project is fine for private sites, but vendors usually use acme_project).
• Templates: single-project.php and archive-project.php.

A full worked example with a taxonomy is in the DEV3 snippet library.