I am running WordPress 4.5.2, Twentysixteen, and a plugin called 'Custom Fields Template'. I am looking to make the most of WordPress' custom fields but do not know how to make it do more than display a UL with some basic text data.
I would like to know how to:
Since you don't provide enough specifics I can only write a very generic answer.
the_meta() you found does not belong to the theme, it is provided by the WordPress core and as you already noticed is not very customizable, it's output can only be styled with CSS.
I assume this is the plugin you are using: https://wordpress.org/plugins/custom-field-template/
Based on that assumption you have these possibilities:
The plugin provides the shortcode
[cft] you can use in the content of the post itself. It's usage is described a little bit in the plugin description, but not much.
If know that you want to show this on every post or every certain post type you can create a template file for this and call the shortcode directly using
echo do_shortcode( "[cft]" );
If you don't like the output of that shortcode either, you can address the custom fields directly.
It looks like the plugin doesn't show you the field names it actually uses, so you need to find them yourself. You can do this either by looking at the database, you'll find them in the table
wp_postmeta, or you add some debug output to your template file, like this:
$meta = get_post_custom( $post->ID ); var_dump( $meta );
This will show you all custom fields from the current post. You can then either use this array to print your values, or you use
get_post_meta() to retrieve specific field values.
Last, but not least: You stated that you are using the default theme Twentysixteen. You should NOT edit any of these theme files, or the files from the custom field plugin. The next time they are updated all your changes are going to be lost. Instead, create a child theme and add your post templates there.