Best Practices Way to Edit Magento Top-Links

In any case do not edit the template file (links.phtml). This serves as a general template for all link lists. For example it is used for the links in the footer also.
With toplinks.php you can do whatever you want because is deprecated since CE v1.4.0.1.
I recommend using the xml files that add the links to the top container to achieve what you need.
The addLink method that is called when adding a new link supports some parameters that allow you to add classes and other attributes on li and a tags in the links and some text before the link and after the link.

public function addLink($label, $url='', $title='', $prepare=false, $urlParams=array(),
        $position=null, $liParams=null, $aParams=null, $beforeText='', $afterText='')
    { ... }

If you want to add an icon to my account menu you can set the $beforeText to <span class="icon"></span> and add some styles on the icon class.
For the cart and checkout links it's a little trickier because they are added via a block object not directly from the layout file.
If you need something different for the cart or checkout lins all you have to do is override the methods Mage_Checkout_Block_Links::addCartLink() or Mage_Checkout_Block_Links::addCheckoutLink(). These 2 call the same addLink() and you can pass different parameters to it.

In case you need to apply custom template only for Top Links, you can do it like this in your theme's local.xml:

<default>
    <reference name="top.links">

        <action method="setTemplate">
            <template>page/template/my_links.phtml</template>
        </action>

    </reference>
<default>

Then copy page/template/links.phtml and rename it to page/template/my_links.phtml and do whatever you need inside that new template file.

To add a custom link to Top Links via local.xml:

<reference name="top.links">
    <action method="addLink" translate="label title">
        <label>My Link</label>
        <url>path/to/page</url>
        <title>My link tooltip</title>
        <prepare>true</prepare>
        <urlParams/>
        <position>150</position>
        <liParams>id="my-custom-id"</liParams>
    </action>
</reference>

Also see this page: http://www.classyllama.com/development/magento-development/editing-magentos-top-links-the-better-way