How to get child element by ID in JavaScript?

Here is a pure JavaScript solution (without jQuery)

var _Utils = function ()
{
    this.findChildById = function (element, childID, isSearchInnerDescendant) // isSearchInnerDescendant <= true for search in inner childern 
    {
        var retElement = null;
        var lstChildren = isSearchInnerDescendant ? Utils.getAllDescendant(element) : element.childNodes;

        for (var i = 0; i < lstChildren.length; i++)
        {
            if (lstChildren[i].id == childID)
            {
                retElement = lstChildren[i];
                break;
            }
        }

        return retElement;
    }

    this.getAllDescendant = function (element, lstChildrenNodes)
    {
        lstChildrenNodes = lstChildrenNodes ? lstChildrenNodes : [];

        var lstChildren = element.childNodes;

        for (var i = 0; i < lstChildren.length; i++) 
        {
            if (lstChildren[i].nodeType == 1) // 1 is 'ELEMENT_NODE'
            {
                lstChildrenNodes.push(lstChildren[i]);
                lstChildrenNodes = Utils.getAllDescendant(lstChildren[i], lstChildrenNodes);
            }
        }

        return lstChildrenNodes;
    }        
}
var Utils = new _Utils;

Example of use:

var myDiv = document.createElement("div");
myDiv.innerHTML = "<table id='tableToolbar'>" +
                        "<tr>" +
                            "<td>" +
                                "<div id='divIdToSearch'>" +
                                "</div>" +
                            "</td>" +
                        "</tr>" +
                    "</table>";

var divToSearch = Utils.findChildById(myDiv, "divIdToSearch", true);

If jQuery is okay, you can use find(). It's basically equivalent to the way you are doing it right now.

$('#note').find('#textid');

You can also use jQuery selectors to basically achieve the same thing:

$('#note #textid');

Using these methods to get something that already has an ID is kind of strange, but I'm supplying these assuming it's not really how you plan on using it.

On a side note, you should know ID's should be unique in your webpage. If you plan on having multiple elements with the same "ID" consider using a specific class name.

Update 2020.03.10

It's a breeze to use native JS for this:

document.querySelector('#note #textid');

If you want to first find #note then #textid you have to check the first querySelector result. If it fails to match, chaining is no longer possible :(

var parent = document.querySelector('#note');
var child = parent ? parent.querySelector('#textid') : null;

(Dwell in atom)

<div id="note">

   <textarea id="textid" class="textclass">Text</textarea>

</div>

<script type="text/javascript">

   var note = document.getElementById('textid').value;

   alert(note);

</script>