How to attach a file with front-end post submission form?

I’m trying to make an assignment uploader for WordPress, and I want users that are logged in to be able to submit data through a form on the front-end and have it create a post in the back-end. I’ve been able to create a shortcode function that allows an admin to choose where they want this form to appear, and when the form submits, it does create a post in the back-end. The problem is, I want the logged in user to be able to attach a file to the post as well, so that the admin can download the file from the back-end.

Here is the shortcode for the form:

add_shortcode('assignmentForm','wprmAssignmentForm');
function wprmAssignmentForm() { 

if (is_user_logged_in()){ ?>
    <form id="custom-post-type" name="custom-post-type" method="post" action="">
        <p>
        <label for="title">Assignment Title</label><br />
        <input type="text" id="title" value="" tabindex="1" size="20" name="title" />
        </p>

        <p>
        <label for="description">Assignment Content</label><br />
        <?php //wp_editor( '', 'description', array( 'media_buttons' => false, 'textarea_rows'=>5 ) ); ?>
        <textarea id="description" tabindex="3" name="description" cols="50" rows="6"></textarea>
        </p>

        <p>
        <label for="upload">Upload File</label><br />
        <input type="file" name="upload" id="upload">
        </p>

        <p align="right"><input type="submit" value="Publish" tabindex="6" id="submit" name="submit" /></p>

        <input type="hidden" name="post-type" id="post-type" value="custom_posts" />
        <input type="hidden" name="action" value="custom_posts" />

        <?php wp_nonce_field( 'name_of_my_action','name_of_nonce_field' ); ?>

    </form>

    <?php

    if($_POST){
        wprmSaveSubmission();
    }

}
else {
    echo '<p>You must be logged in to submit an assignment!</p>';
}

}

And here is the saving function:

function wprmSaveSubmission() {

if ( empty($_POST) || !wp_verify_nonce($_POST['name_of_nonce_field'],'name_of_my_action') ) {
    exit;
}
else {

    // Basic validation
    if (isset ($_POST['title'])) {
        $title =  $_POST['title'];
    } 
    else {
        echo 'Please enter a title';
        exit;
    }
    if (isset ($_POST['description'])) {
        $description = $_POST['description'];
    }
    else {
        echo 'Please enter the content';
        exit;
    }

    // Add the content of the form to $post as an array
    $post = array(
    'post_title' => wp_strip_all_tags( $title ),
    'post_content' => $description,
    'post_status' => 'publish',            
    'post_type' => 'wprm-assignments'  
    );
    wp_insert_post($post);  

}

}

I have the file uploader field in the form, but I don’t know how I can have the upload saved somewhere and then be offered as a download with the post in the backend. Any ideas on how I could achieve this?