How to get custom field value in frontend for taxonomy

I would suggest to use update_term_meta() and get_term_meta() as they were introduced in WordPress 4.4. This would help to keep your wp_options table smaller.

But either way, you need to know the ID of the term in the “Frontend”.

Using Term meta you would need it:

$term_meta = get_term_meta( $term_id, 'series_images', true );

Using your solution you would need it:

$term_meta = get_option( "weekend-series_" . $t_id );

So the question is:

How do I get the ID of a term?

How to get the current term ID on the taxonomies page

A very useful function is get_queried_object(), which returns the queried object. If you are in the taxonomy.php template, or the tag.php or the category.php this would be the current term object:

WP_Term Object
(
    [term_id] => 20
    [name] => Schlagwort
    [slug] => schlagwort
    [term_group] => 0
    [term_taxonomy_id] => 20
    [taxonomy] => post_tag
    How to get custom field value in frontend for taxonomy => 
    [parent] => 0
    [count] => 0
    [filter] => raw
)

So to get the terms ID in these templates you can do something like:

$current_object = get_queried_object();
$term_id = $current_object->term_id;

How do I get the term IDs of terms attached to a certain post?

If you want to display those images lets say in the single.php template, you would need to get the term IDs attached to the current post. With get_the_terms() you can get exactly those. If you use it inside the loop* you can simply do something like this:

$terms = get_the_terms( get_the_ID(), 'post_tag' );
foreach ( $terms as $term ) {
    $term_id = $term->term_id;
    /* Do something with the $term_id */
}

The first parameter is the current post ID while the second parameter is the slug of the taxonomy (in my example its post_tag for the tags). What you get in return is an array of Term Objects.

How to get the term IDs of all terms of a taxonomy

At last, lets say, you have a taxonomy and you want the term IDs of all the terms attached to this taxonomy get_terms() is your friend.

$terms = get_terms( 'post_tag' );
foreach ( $terms as $term ) {
    $term_id = $term->term_id;
    /* Do something with the $term_id */
}