How to pull a number posts from a specific Taxonomy within a Custom Post type

The get_the_terms() returns a list of term objects. So first of all, you need to treat the output as a list rather than a single element. The name you’re looking for is stored in the name field.

$artistnames = get_the_terms($post->ID, 'artist');
if (!empty($artistnames)) : ?>
    foreach ($artistnames as $obj) {
         $name = $obj->name;
         // Do something with the name
    }

Maybe you should use a custom field instead of a taxonomy? My experience is that they work better for singletons.

You get the custom fields in the loop by get_post_meta():

<?php $meta_values = get_post_meta($post->ID, 'artist', true); ?> 

Hopefully this helps you somewhat!