Savings settings in a custom PHP plugin for WordPress

First, I’d like to pre-emptively apologize for doing anything naive or foolish. I’m a novice programmer, but I’m trying to learn. Right now, I’m trying to develop a plugin for wordpress as part of my internship. The plugin does what it needs to do for the most part, but I just can’t seem to make the settings page work the way I want. I know the initialization part of the plugin is working properly, but I’m having trouble with using the settings page. Specifically, I don’t where or how to use _POST in order to save the information. I also don’t where to place the update_option function call after I’ve received the information from the settings page. Here’s my code right now:

add_action( 'admin_menu', 'menu' );
add_action( 'admin_init', 'plugin_admin_init' );

function plugin_admin_init() { // whitelist options
  register_setting( 'plugin_options', 'plugin_options', 'plugin_options_validate' );
  add_settings_section('plugin_main', 'Main Settings', 'plugin_section_text',           'plugin');
  add_settings_field('plugin_text_string', 'REE View Count Settings',           'plugin_setting_string', 'plugin', 'plugin_main');
}

function menu() {
    add_options_page( 'REE View Count Options', 'REE View Count', 'manage_options',     'plugin', 'plugin_options_page' );
}

// generates the settings web page
function plugin_options_page() {
    echo '<div class="wrap">';
    echo '<h2>Options</h2>';
    echo '<form action="' . $_SERVER['PHP_SELF'] . '" method="post">'; 
    // In the line above, I don't know what to put. The file name of the plugin?
    // The URL of the settings page on the admin page? Or what I have currently? 
    // Both the first and last options bring up a 404 page.
    settings_fields('plugin_options');
    do_settings_sections('plugin');
    echo '<input name="Submit" type="submit" value="' . esc_attr_e('Save Changes') . '"         />';
    echo '</form>';
    //update_option('plugin_options', $_POST['plugin_options']);
    // Where should I try to update the option? And how?
    echo '</div>';
}

//from add_settings_section parameter 3, in plugin_admin_init
function plugin_section_text() {
    echo '<p>Main description of this section here.</p>';
}

//Function from 3 paramter of add_settings_field in plugin_admin_init that outputs     input HTML
function plugin_setting_string() {
    $options = get_option('plugin_options');
    echo 'value of options is: ' . $options;
    echo "<input id='plugin_text_string' name='plugin_options' size='40' type='text' 
    value='{$options}' />";
}

//Validation function from register_setting in plugin_admin_init
function plugin_options_validate($input) {
    echo "this is input: " . $input;
    $newinput = trim($input);
    if(!preg_match('/^[0-9]+$/', $newinput)) {
        $newinput = 10;
    }
    update_option('plugin_options', $newinput);
    return $newinput;
}

Note: I don’t actually use these exact function/variable names – I have them prefixed with the actual plugin name.

Can you help me? I want to take the information the admin inputs into the settings page and update the database with the that information (which in this case is just a single number). I understand that I need to use PHP _POST, but I don’t know how. Furthermore, I don’t where to post to, because when I use action=”the_file_name.php,” I get a 404 error upon submission. What should the action of the form be so that I can use the information that I got from the admin submission and use it for later? And after I do that, how do update the setting? And by how, I mean where do I place update_option? I apologize if this seems rambly or vague – I feel like I’m somewhat over my head.

If it helps, I’ve been building this settings page in the same file as the plugin itself, with the help of this guide: http://ottopress.com/2009/wordpress-settings-api-tutorial/

Unfortunately, I don’t see anything in that guide that speaks of updating information, just creating the page itself.