Use Ajax on WooCommerce Login & Registration Forms Validation.

The Ajax login/register is a free light-weight plugin for WooCommerce that changes the login, registration and lost password forms to use Ajax and jQuery validation to avoid the standard WooCommerce validation process that loads the page to display error messages and to provide your customers with improved user experience.

The plugin settings (WooCommerce > Settings > Accounts & Privacy allow you to set the password strength level you will require from your customers and to modify the password error messages.

If can redirect users after successful login or registration using these code snippets.