How do I list a custom field and custom taxonomies for each result in a loop?

You can get custom field values with get_post_meta. Like so…

<?php
$wp_query = new WP_Query();
$wp_query->query('post_type=customtype&showposts=10'.'&paged='.$paged);
while ($wp_query->have_posts()) : $wp_query->the_post();

the_title(); // <- works
if( $some_meta = get_post_meta( get_the_ID(), 'the_meta_key_you_want', true )
{
    // you could put other formatting here.
    echo $some_meta; 
}

endwhile;

You’ll need to change the_meta_key_you_want in the call to get_post_meta, of course.

For a custom taxonomy, you’ll want to use get_the_terms or the_terms. the_terms is a bit more like the_tags or the_category, it spits out HTML for you. get_the_terms will just return an array (a list) of the term objects.

Here’s an example of get_the_terms. The Codex entry for it also has some good examples. You’ll have to change your_taxonomy, of course.

<?php
$wp_query = new WP_Query();
$wp_query->query('post_type=customtype&showposts=10'.'&paged='.$paged);
while ($wp_query->have_posts()) : $wp_query->the_post();

the_title(); // <- works
$terms = get_the_terms( get_the_ID(), 'your_taxonomy' );
if( $terms && ! is_wp_error( $terms ) )
{
    foreach( $terms as $t )
    {
        echo '<a href="' . get_term_link( $t->slug, 'your_taxonomy' ) . '">' . esc_html( $t->name ) . '</a>';
    }
}

endwhile;