Redirect /member/ to /member/user

You could hook into template_redirect action hook and redirect the users. I assume that you want to redirect the currently logged in user.

add_action( 'template_redirect', 'wpse314345_redirect_users' );
function wpse314345_redirect_users( $template ){
    // Check if the user is logged in
    if( is_user_logged_in() ) { 
        // Get the current user
        $user = wp_get_current_user();
        // Redirect the user
        wp_safe_redirect( esc_url( site_url( 'member/' . $user->display_name) ) );
        exit();
    }

    return $template;
}

You can use other action hooks too, but make sure the action isn’t too soon ( query not set up yet ) or too late ( header already sent ).

However this will redirect every request to that particular URL. You might want to also do additional checks, based on your needs.