How to select nested DOM elements inside 'ul' element

I'm not sure if you want to limit the action somehow, but getting all anchor elements in the page is easy:

var links = document.getElementsByTagName('a');

If you want to limit your search inside an element, set an id on that element so that you can easily find it, and use the getElementsByTagName on the element:

var links = document.getElementById('menu').getElementsByTagName('a');

$$('.menu a')

// for the links of the first ul.menu on the page
var menuLinks = document.getElement("ul.menu").getElements("a");
// or you can get all links children of all uls with class menu
var menuLinks = document.getElements("ul.menu a");