Adding padding to icon in header.php

This is best done with CSS. If you want each icon used to have unique CSS, wrap it in a div like this:

<div class="sm_icon_1"><?php echo do_shortcode('[social_links facebook="test"]'); ?></div>

Example CSS:

.sm_icon_1 {
padding: 10px;
margin:10px;
}

Forum users cannot be more specific without a link to your site.

Then add the class(es) and style them adding padding, margin, etc.

To discover what CSS is output by your theme and to add new styles, start with a a web inspection tool such as Firebug: http://getfirebug.com/ , Chrome Inspect Element: https://developers.google.com/chrome-developer-tools/ or Internet Explorer 9+ F12 Dev. Tools: http://msdn.microsoft.com/en-us/library/dd565627(v=vs.85).aspx#csstool

There are others.

When editing CSS, use a Child Theme
http://codex.wordpress.org/Child_Themes
Custom CSS Plugin, or Theme provided custom CSS option.
Edits to parent themes are lost on theme update.

Learn CSS: http://www.w3schools.com/css/