Browse the store and configure a product (billing cycle, options, a domain if the product needs one) without signing in. Nothing asks you to register while you shop.
Your account is created at the payment step. The checkout page asks for your name, email and a password right above the payment method. Press the order button once and the account is opened, you are signed in and the order is placed.
Already have an account? The checkout page has a Sign in link. Log in there and your cart comes with you.
After ordering you land in the client area, where the order and its invoice are waiting under Services and Billing → Invoices.