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;