Find td with specific text, and operate on the td right after that one?

This should work...

$(function() {
    var searchText = "Section Heading 3";
    var output = "";
    $("td").each(function(i, item) {
        if($(item).html() == searchText) {
            output = $(item).html();
            output += $(item).closest("td").html()
        }
    });

    console.log(output);
});

Here's a jsFiddle http://jsfiddle.net/5ELLM/

jQuery has :contains() pseudo class :

$('td:contains("Section Heading 4")').next().text(function(_,t){
    return t + ' Hello';
})

Be carefull, :contains is case sensitive.

First, you want to use :contains:

jQuery( "td:contains(text)" );

Then, you can use .next and .append:

jQuery( "td:contains(text)" ).next().append("Hello");

Here's a working snippet to demonstrate usage:

jQuery(function($) {
  $("td:contains(Text to Find)").next().append(" Hello");
});

table {
  border-collapse: collapse;
}

td {
  border: 1px solid #ccc;
  padding: 2px 5px;
}

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.2.4/jquery.min.js"></script>
<table>
  <tbody>
    <tr>
      <td>Some Text</td>
      <td>More Text</td>
      <td>YMT</td>
    </tr>
    <tr>
      <td>Text to Find</td>
      <td>Modify:</td>
      <td>Do nothing here</td>
    </tr>
  </tbody>
</table>

My initial answer missed the point about updating the text, here is a sample with this included:

$(".test td:contains(Heading 1)")
    .next().text(function(){
        return $(this).text() + " Hello"
    });

I don't believe you can use the append method as I think it only works to add html tags.