I've added the "recent posts" widget, but I was hoping to customize the look surrounding it, but I have no idea where that kind of stuff is located.
I'm sure it's possible, and I don't think every theme I've seen uses the same style for each widget, but I can't seem to find out which file contains the code I need to edit.
I use this widget on my site too. Depending on what you need to do, There are two places that you modify code to customize the widget:
1. Your theme's 'style.css' file
The following id and classes are applied to the <li> tag that wraps around the widget.
#recent-posts
.widget
.widget_recent_entries
2. Wordpress Core
I believe that the core Wordpress function that generates the html can be found here:
/wp-includes/widgets.php -> line 806