How do you know if the current element is not the first-child?
It should work with $(this), for example:
$(\"li\").click(function(e) {
          
        For all li's in #thing that aren't first child:
$('#thing li:not(:first-child)')
see http://api.jquery.com/not-selector/
Also, you can check if $(this).prev().length is defined, like in:
if (!$(this).prev().length){ //This means $(this is the first child
    //do stuff
}
For people who want a vanilla JS solution:
<div id="parent">
  <div id="child-1"></div>
  <div id="child-2"></div>
  <div id="child-3"></div>
</div>
let element = document.getElementById('child-1');
console.log(element.parentElement.firstElementChild === element); // true
console.log(element.parentElement.lastElementChild === element);  // false
element = document.getElementById('child-2');
console.log(element.parentElement.firstElementChild === element); // false
console.log(element.parentElement.lastElementChild === element);  // false
element = document.getElementById('child-3');
console.log(element.parentElement.firstElementChild === element); // false
console.log(element.parentElement.lastElementChild === element);  // true
It's not the most elegant thing but it gets the job done.
You can even check if it's the nth element in the same fashion.
let n = 2; // check if it's the 2nd child
let element = document.getElementById('child-2');
// index starts at 0
console.log(element.parentElement.children[-1 + n] === element); // true
jQuery .not()
$(this).not(':first');
If you would like to select everything except the first child, this is how to do it:
$('#some-parent').children().not(':first')
$("li").click(function(e) {
   if ($(this).index != 0 )
  {
      /* do something */
   }
});