There is a default option in WooCommerce from where you can control "Add to cart" redirection whenever a user adds a new product to their cart.

You can find the option in the WooCommerce -> Settings -> click on the Products tab.

When the option “Redirect to the cart page after successful addition” is checked, it will redirect all users to the cart after adding a product to the cart. If unchecked, the page will not get redirected.

This is how you can redirect the users to the Cart page after they've added a new product to their cart.