Creating login for client / customer that will take them to customized part of site

There is a individual personal page, the user profile. I don’t know about the functionality you are talking about, but you can use the login_redirect hook to redirect users after logging in. To get the URL needed we use the get_edit_profile_url(), which returns the link to before mentioned user profile.

add_action( 'login_redirect', 'wpse194827_redirect_to_user_profile', 10, 3 );
function wpse194827_redirect_to_user_profile( $redirect_to, $request, $user ) {
  return get_edit_profile_url();
}