Display search result link direct to image with caption?

That depends. Do the images exist as individual posts in the wp_posts database? Are they in the media library? I don’t know how Envira Galleries works, so I can’t say for sure, but if the images exist as individual posts in the media library, then there should be no problems in having the search find them: just index the attachment post type and include that in the search.

When the search finds an attachment page, the default setting is to point to the attachment page (not the post or page that has the attachment, but the WP-generated page for that single image). That can be modified in the search results template so that the search result points directly to the image.

Currently, searching for the attachment post type on your page (by adding &post_types=attachment at the end of the search page URL) finds nothing, so to me it seems like the attachments aren’t even indexed. Can you show me a search that does find a post of attachment post type?

Envira probably stores the caption somewhere in its own files. In that case it’s probably possible to index it, but it will be more complicated: you’d have to write some code that finds the image in the Envira database and gets the caption from there. But that gets a bit complicated.

Envira stores the image title in the gallery post meta data. Here’s a function that goes through all gallery post meta data and digs the image titles, alt texts and captions from there:

add_filter( 'relevanssi_content_to_index', 'rlv_envira_metadata', 10, 2 );
function rlv_envira_metadata( $content, $post ) {
	global $wpdb;

	$gallery_data = $wpdb->get_col(
		$wpdb->prepare(
			"SELECT meta_value FROM $wpdb->postmeta AS pm, $wpdb->posts AS p WHERE p.ID = pm.post_id AND p.post_type = %s AND meta_key = %s",
			"envira", "_eg_gallery_data"
		)
	);
	foreach ( $gallery_data as $data ) {
		$gallery = unserialize( $data );
		if ( isset( $gallery['gallery'][ $post->ID ] ) ) {
			$photo_content[] = $gallery['gallery'][ $post->ID ]['title'];
			$photo_content[] = $gallery['gallery'][ $post->ID ]['alt'];
			$photo_content[] = $gallery['gallery'][ $post->ID ]['caption'];
			$content .= ' ' . implode( ' ', $photo_content );
		}
	}
	return $content;
}

If you add this to your theme functions.php and make Relevanssi index the attachment post type, you should be able to find posts by their titles and captions.

Yes, I know there’s a difference between Envira Gallery and the Media Library. That’s tricky, especially if the same image appears in multiple galleries with different captions; if the images appear only in one gallery each, getting reasonable search result pages is possible – a similar method can be used to dig in the titles and captions for excerpts.

Adding this should help a little with the excerpts:

add_filter( 'relevanssi_excerpt_content', 'rlv_envira_metadata_excerpts', 10, 2 );
function rlv_envira_metadata_excerpts( $content, $post ) {
	global $wpdb;

	$gallery_data = $wpdb->get_col(
		$wpdb->prepare(
			"SELECT meta_value FROM $wpdb->postmeta AS pm, $wpdb->posts AS p WHERE p.ID = pm.post_id AND p.post_type = %s AND meta_key = %s",
			"envira", "_eg_gallery_data"
		)
	);
	foreach ( $gallery_data as $data ) {
		var_dump($data);
		$gallery = unserialize( $data );
		if ( isset( $gallery['gallery'][ $post->ID ] ) ) {
			$photo_content[] = $gallery['gallery'][ $post->ID ]['caption'];
			$content .= ' ' . implode( ' ', $photo_content );
		}
	}

	return $content;
}

This will make Relevanssi use the captions for excerpts. To get the title showing is a bit more complicated and requires changing your search results templates – another approach is to just give the photos good titles in the Media Library. Or you could create a plugin that copies the Envira captions and titles to Media Library, that too would sort this out.