Not really confident that techniques may be the correct terminology...

This is actually the original working code:

<a href="<?php bloginfo('url'); ?>">
<img src="<?php bloginfo('stylesheet_directory'); ?>/images/logo.png" alt="Polished Logo" id="logo"/></a>
<img src="<?php bloginfo('stylesheet_directory'); ?>/images/separator.png" width="2" height="59" alt="Line" class="logo_line"/>
<p id="logo_title"><?php bloginfo('description'); ?></p>

I needed it to simply execute around the home page, and so i authored this:

if ( $_SERVER["REQUEST_URI"] == '/' ){
echo '<a href="'.bloginfo('url').'">
<img src="'.bloginfo('stylesheet_directory').'/images/logo.png" alt="Polished Logo" id="logo"/></a>
<img src="'.bloginfo('stylesheet_directory').'/images/separator.png" width="2" height="59" alt="Line" class="logo_line"/>
<p id="logo_title">'.bloginfo('description').'</p>';

However it results the bloginfo() and also the other terms completely outdoors the html tags I've produced. For example, with bloginfo('stylesheet_directory') it'll display your directory outdoors the IMG tags I produced.

Any ideas? Apparently my syntax is not correct or something like that....

bloginfo() results data with echo and returns nothing, so rather than attempting to concatenate everything, just output in sequence, e.g.

echo '<a href="';
echo '"><img src="';

Ugly I understand, but see answer by Nithesh for any possible alternative.

bloginfo function directly echoes the output. Within this situation you need to use get_bloginfo to include the came back value towards the string and echo the entire string. In my opinion this works

if ( $_SERVER["REQUEST_URI"] == '/' ) {
  echo '<a href="'.get_bloginfo('url').'">
    <img src="'.get_bloginfo('stylesheet_directory').'/images/logo.png" alt="Polished Logo" id="logo"/></a>
    <img src="'.get_bloginfo('stylesheet_directory').'/images/separator.png" width="2" height="59" alt="Line" class="logo_line"/>
    <p id="logo_title">'.get_bloginfo('description').'</p>';

This is a better alternative:

<?php if ( $_SERVER["REQUEST_URI"] == '/' ) { ?>
<a href="<?php bloginfo('url') ?>">
  <img src="<?php bloginfo('stylesheet_directory') ?>/images/logo.png" alt="Polished Logo" id="logo"/>
<img src="<?php bloginfo('stylesheet_directory') ?>/images/separator.png" width="2" height="59" alt="Line" class="logo_line"/>
<p id="logo_title"><?php bloginfo('description') ?></p>
<?php } ?>

I additionally suggest while using is_home() function supplied by wordpress to check on for that home page rather than checking the $_SERVER['REQUEST_URI'] value.