WordpPress | Get the first term of a taxonomy assigned to a post

first_term.php

<?php
$taxonomy = 'taxonomy';
if( has_term( '', $taxonomy ) ) {
    $terms = get_the_terms( $post->ID, $taxonomy );
    $term = array_shift( $terms );
    $term_link = get_term_link( $term->term_id );
    $term_name = $term->name;
}
?>

First term from the taxonomy terms assigned to a post.

Note this is different than get_term which simply, well, gets the term, this gets the terms assigned to the post using get_the_terms.

There are different ways to get the first element from an array, here we use the array_shift function. Also see array_pop which gets the last element.