单独的表单，按特定顺序执行2个操作

I've got a form asking for user input. The input should be a name which exists in my database.

<p><span class="error">* required field.</span></p>
<form method="post" action="<?php echo htmlspecialchars($_SERVER["PHP_SELF"]);?>">
    Name: <input type="text" name="name" value="<?php echo $name;?>">
    <span class="error">* <?php echo $nameErr;?></span>
    <br><br>
    <input type="submit" name="login" value="Login">
</form>

I'm trying to first check user input, and then if the input is clean, send the user off to another site.

This is what I'm using to scrub the input:

if ($_SERVER["REQUEST_METHOD"] == "POST") {
    if (empty($_POST["name"])) {
        $nameErr = "Name is required";
    } else {
        $name = test_input($_POST["name"]);
        // check if name only contains letters and whitespace
        if (!preg_match("/^[a-zA-Z ]*$/",$name)) {
            $nameErr = "Only letters and white space allowed"; 
        }
    }
}

Then I would like to send to the next site after querying to my database and receiving an ID:

<form action = "http://website.com/~username/site.php" method="get">
    Name: <input type="text" name="name"><br>
    <Input type="submit">
</form>

How should I go about doing this?

Use header() function to achieve your goal

header('Location: yourfile.php?id=' . $yourIdValue); // adopt to your needs

In your target script fetch id with

$id = $_GET['id'];