How to automatically check multiple categories on new post

The filter you’re looking for is called wp_terms_checklist_args and is used by wp_terms_checklist() (which is internally called to display the category metabox).

You can use this to pre-selected specific categories (by ID) under certain circumstances as shown in this example

add_filter('wp_terms_checklist_args', 'WPSE_pre_select_categories', 10, 2);
function WPSE_pre_select_categories($args, $post_id) {
        $post = get_post($post_id);

        // only pre select categories for new posts
        if ($post->post_status !== 'auto-draft' || $post->post_type !== 'post')
                return $args;

        // select categories with ID 4 and 6
        $select_categories = [4, 6];

        // little hack so array_merge() works if default is NULL
        if (empty($args['selected_cats'])) {
                $args['selected_cats'] = [];
        }

        // array_merge() with numerical indices only appends
        // so I use array_unique() to remove any duplicates
        $args['selected_cats'] = array_unique(array_merge($args['selected_cats'], $select_categories));
        return $args;
}