Display categories and their IDs

Are you looking for a list of the current post’s categories and parents, or just a general list of categories? If the former, you can use get_category_parents(). If you’re in The Loop, for example:

<?php
    if( have_posts() ) : 
        while( have_posts() ) :
            the_post();
            // Post stuff
            // ...

            $cats = wp_get_post_categories( $post->ID );
            foreach( $cats as $c ) {

                $category_list=""; // initialize our text strings
                $category_id_list="";

                $display = false;
                $sep = ', ';
                $nice = false;
                $parents = get_category_parents( $c, $display, $sep, $nice );

                $_parents = explode( ',', $parents ); // convert the string to an array
                $_parent_ids = array(); // initialize the array for the IDs
                foreach( $_parents as $p ) {
                    $_parent_ids[] = get_cat_ID( $p );
                }
                // convert the $_parent_ids array to a string
                $parent_ids = implode( ', ', $_parent_ids );
                echo( "$parents $parent_ids" );
            }

        endwhile;
    endif;
?>

[Edited: Cleaned up the code, and took advantage of PHP’s implode() function]