How do I retrieve a PHP variable in my HTML code?

I'm not sure what you mean by "not breaking it again with a PHP tag". HTML on its own cannot access PHP variables. This is because PHP is a server-side language and HTML is a client side language. But here is the simplest way to print php variables retrieved from a form.

<form>
    <p>Your name: <?php echo $_POST['name'] ?></p>
</form>

This is assuming that there was a form that submitted a field called 'name' using POST.

You can process the name in a php script at the top of the file and then simply echo it when you're printing the html. This way, you won't have too much php code mixed in with the HTML (which makes it look cleaner).

Once you got the values in the PHP script, are you calling a new script? If so, you might wanna save the values in $_SESSION["varible_name"]. If not, you just have to echo it.

It depends on how you are accessing your form data, either through $_POST or through $_GET. For simplicity, I'll assume your using $_GET and modify this example for more clarity.

So lets say you have a form hosted on welcome.php:

<form action="welcome.php" method="get">
Name: <input type="text" name="fname" />
Age: <input type="text" name="age" />
<input type="submit" />
</form>

Now the results will be returned back to the same page, so you want to modify the page to say:

<form action="welcome.php" method="get">
Name: <input type="text" name="fname" value="<?php echo $_GET["fname"]; ?>"/>
Age: <input type="text" name="age" value="<?php echo $_GET["age"]; ?>" />
<input type="submit" />
</form>

Though you'll notice that we're using the same page, and we can only have one version of the page, so we want to render if upon the condition that our variable has been set.

if (isset($_GET["fname"])){
//code to print second form
}
else{
//code to print first form
}

Or, in another way (using the ternary operator):

<form action="welcome.php" method="get">
Name: <input type="text" name="fname" value="<?php echo ((isset($_GET["fname"]))?$_GET["fname"]:""); ?>"/>
Age: <input type="text" name="age" value="<?php echo ((isset($_GET["age"]))?$_GET["age"]:""); ?>" />
<input type="submit" />
</form>