The sum bicycle plus shopping is seen as something difficult or even incompatible by non-bikers sometimes. But when it comes to use your brain to solve this apparently important problem, it all has sense.
First of all, freedom is increased in comparison to using a car. You can park your bike easily and it occupies less space. Secondly, you exercise your body.
Moreover, you need a rear rack, saddlebags or just a backpack. Regarding saddlebags, the best way to organize them is putting the heavier and less fragile objects at the bottom, i.e. tins and boxes. Next, put the most breakable items on top such as eggs and bread. Remember to distribute weights uniformly in your saddlebags in order to avoid your bike bending.
Thirdly, you can use a bicycle trailer or a cargo bike to carry extra shopping weight in addition to people. Finally, never ever forget locking your bicycle by at least two points while going shops.