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How add ID to HTML href with javascript

I have an HTML like this
<div class="this">
EXP
</div>
I want to add id to <a>. But do not know what to do.

First select your element using something like .getElementsByClassName(). Keep in mind that .getElementsByClassName() returns a NodeList collection of elements, so you'll want to access the first index (or loop over them). You can then simply set the ID with .id, as the ID is merely a property of an element.
This can be seen in the following:
const element = document.getElementsByClassName('this')[0];
element.id = 'element';
console.log(element);
<div class="this">
EXP
</div>

If you want to add this with Javascript, you'll need to use a selector to target your <a> tag and then set the id attribute on it. You can do this by using the querySelector() function or as seen below:
// Find an <a> tag that occurs below a class called "this" and set its id attribute
document.querySelector('.this > a').id = "some-id";
There are many other available functions to handle this through native Javascript and other frameworks, so your milage may vary depending on what you are using.
Example
In this example, we have provided some CSS that should only apply to an element with an id of "test" and we'll run the necessary code to show that the id is being added to the element (as it will be red):
document.querySelector('.this > a').id = 'test';
#test { color: red; }
<div class="this">
EXP
</div>

Add the id attribute to the <a> tag. See the differences of the middle line:
<div class="this">
<a id="expid" href="exp.com">EXP</a>
</div>

How to replace all id from href?

I have the following markup
<div class = "general">
<div id ="custom"></div>
</div>
How to change id = "custom" in all <div> with class="general" from href on page using jQuery?

You can try this:
$("div.general").each(function() {
$(this).children("div#custom").text($(this).children("a").attr("href"));
});
If I understand you correctly, you want to iterate through all div.generals, and change the text of each child div#custom to the href of the child a.
See a working example on JSfiddle.
Also, another tip is to avoid using multiple elements with the same id. In your code you have a <div> with id="custom". You also say that the div.general appears multiple times — therefore, the id "custom" will appear multiple times. This is bad practice. I suggest that you change id to class.

You need to loop through all div.general and replace the id attribute of div#custom to whatever is there as the anchors href property. The following code will work:
$(".general").each(function(){
$(this).find("#custom").attr("id", $(this).find("a").attr("href").replace("#", ""));
})
Here the .find() will dig out elements from any depth inside the parent. If you are sure about the DOM position of the elements, you can change the .find() to .children()

How would I get a classes href below it?

<li class="catalog-list-item" data-icon="false">
<a href="/items/170893265">
How would I get the href /items/ and console.log the ID under catalog-list-item?
I tried parentNode and stuff, nothing seems to work for me.

To log the href attribute, I'd do something like this
$(".catalog-list-item a").each(
function(){
console.log($(this).attr("href"));
});
I don't completely catch the idea of logging the ID - there is no ID mentioned in your code example.

By using jQuery selector (jQuery documentation, W3Schools), you can retrieve all anchor tags that are children of an element with the '.catalog-list-item' class.
var allAnchorTags = $(".catalog-list-item a");
You can get the href attribute value by using .attr-method.
With the .replace-method you can replace the '/item/' string in front of the id.

How to read the contents of a href with javascript?

I have the following html and am stumped as to how to read the contents of the href tag?
<p class="xyz-title">
<a href="http://www.youtube.com/embed/xyz">
<span class="field-content">Title here</span>
</a>
</p>
I tried document.getElementByClass('xyz-title')[0].innerHTML but that didn't work obviously.
Thanks very much for pointing me in the right direction.

It is a syntax error. It is supposed to be getElementsByClassName. Use this instead:
document.getElementsByClassName('xyz-title')[0].innerHTML
And for selecting the <a> tag inside the <p class="xyz-title"> You need to use this code:
document.getElementsByClassName('xyz-title')[0].children[0].href
document.getElementsByClassName('xyz-title')[0].getElementsByTagName('a')[0].href
Or, you can simply use:
document.getElementsByTagName('a')[0].href
Fiddle: http://jsfiddle.net/praveenscience/DZhRv/

.innerHTML will give you the content of the element, not the value of any attributes. .href will give you the href value.
You tried to use getElementByClass() by there is no such function - you want getElementsByClassName() (as per the tag that you added to the question), a function that returns a list so you do need the [0] to get the first one.
To select the anchor element, try:
document.getElementsByClassName('xyz-title')[0].getElementsByTagName('a')[0].href
Or, simpler:
document.querySelector('.xyz-title a').href
Demo: http://jsfiddle.net/6Pg43/
Note that either way will give an error if the elements don't exist.

Working FIDDLE Demo
Try this:
var href = document.getElementsByClassName('xyz-title')[0].getElementsByTagName('a')[0].href;
alert(href);

How to target a div with a class and an inner p tag

I have a structure that looks like this..
<div class="infobubble">
<p>
PLACE CONTENT HERE
</p>
</div>
How do I use jquery to target the in tags?
I tried this but did not work.
$("infobubble p").html('My Text');

Your code is wrong, you need to specify that you are looking for a class, like so
$(".infobubble p").html('My Text');
Your code $("infobubble p") would be looking for a tag element named infobubble which does not exist

$(".infobubble p").html('My Text');
in jQuery you need to add an '.' to select by class or '#' to select by id.
If no '.' or '#' is specified, jQuery will try to find an element by tag name... in your case it was trying to find <infobubble> which isn't an element.