If online check-in is not possible, you can always check-in at the airport (note: in some cases there is a surcharge). In all cases, make sure you are at the airport on time because of longer waiting times at check-in.
On the day of departure, you take your luggage to the airport. If you have checked luggage you take it to the designated baggage drop-off point or to the check-in desk. Some airports even have super convenient self-service check-in machines. Then you proceed to security. Your boarding pass will be scanned/checked, so be sure to keep your paper or mobile boarding pass in your hand. In most cases you need to pass through customs to have your passport checked.
If there is any time left, you can still shop at the airport after customs, but make sure you are at the gate on time. The screens at the airport show flight details (gate number and boarding time of the flight). If you miss your flight, your luggage will be offloaded and in most cases you will have to book a new flight ticket.