Till (GB) and cash register (GB/US - but not just 'register) are the actual objects that the money goes into.
A cashier is the person who handles cash, particular in a bank.
As for the expression that a customer would use when shopping, I'd agree with Paddy. These days we tend to use the word 'checkout' for the area of the store where payments are made, and the commonest question for a shopper to ask is simply 'Where can I pay for this?'
Members of staff might use the terms 'till' or 'cash register', but a customer probably wouldn't.