Remove the WordPress logo from Admin Bar:

The following code have to be added in the theme’s functions.php file.

function remove_wp_logo( $wp_admin_bar ) {
$wp_admin_bar->remove_node( 'wp-logo' );

add_action( 'admin_bar_menu', 'remove_wp_logo', 999 );

Remove various admin dashboard widgets:

function remove_wpdashboard_widget() {
//For more functions, check wp-admin/includes/dashboard.php
// grep -inr 'wp_add_dashboard_widget' --color
// core=left side

//Remove WordPress News widget
remove_meta_box( 'dashboard_primary', 'dashboard', 'side' );

//Remove Quick Draft
//remove_meta_box( 'dashboard_quick_press', 'dashboard', 'side' );

//Remove Activity
//remove_meta_box( 'dashboard_activity', 'dashboard', 'normal' );

//Remove at a glance
//remove_meta_box( 'dashboard_right_now', 'dashboard', 'normal' );

// Hook into the 'wp_dashboard_setup' action to register our function
add_action('wp_dashboard_setup', 'remove_wpdashboard_widget' );