Where to insert customize_register in wordpress plugin

You can use the customize_register action like this:

function my_customize_register( $wp_customize ) {
    /* Just use the $wp_customize object and create a section or use a built-in
       section. */
    $wp_customize->add_section(
        'my_section',
        array(
            'title'       => 'My Section',
            'priority'    => 30,
        )
    );
    /* Now you can add your settings ... */
    $wp_customize->add_setting(
        'my_options[my_first_option]',
        array(
            'default'    => '',
            'type'       => 'option',
            'capability' => 'edit_theme_options',
        )
    );
    /* ... and link controls to these settings. */
    $wp_customize->add_control(
        'my_first_option',
        array(
            'label'      => 'My First Option',
            'section'    => 'my_section',
            'settings'   => 'my_options[my_first_option]',
        )
    );

}
add_action( 'customize_register' , 'my_customize_register' );

You can use built-in sections or define your custom ones. Define your settings as you like and link controls to them.

The page Theme Customization API has a lot of useful code … but, because you asked for tips: Always set WP_DEBUG when you develop code for WordPress. You’ll see very often immediately the reason why something is not working as expected.