I have a list of links stored in a Sharepoint List, and have them displayed in the standard list-item-displaying web part. I would like to have this web part formatted (perhaps with CSS) in a custom manner, while still maintaining the link to the underlying Sharepoint List. I understand that I could achieve this goal by creating a web part from scratch, but I am wondering if there is a way to get this done through the out of the box web part?
You have different ways to do this.
The cleanest one would be to modify the xslt attached to your Web Part. This can be done via the user interface in SP 2010, in SP 2007 you would need SharePoint Designer. You'll be able to apply simple formattng or conditional formatting.
If the formatting is part of your page/site branding, then Ryan's suggestion works just fine. You can include additional css on a specific page via a Content Editor Web Part, or add a custom stylesheet to your Master page.
You can absolutely apply branding to OOB links lists. Use IE developer tools or Firefox+Firebug to select the link elements, this will give you the CSS classes used - you can then apply these classes in your own stylesheet with your own styles.