For starters, keeping certain things in mind will be beneficial. Firstly, you must know how to ride a bike. If you have not used the bike for a long time, then a test ride will be good. Also check the condition of the tires and whether they need replacement.
Now get started. Find a grocery store near your home and outline an easy route which you can use. Avoid entering busy roadways, but don't use bumpy shortcuts either, where maintaining balance could be difficult.
Buy yourself a good backpack, which will come in extremely handy during shopping. Look for a large sized bag, with plenty of storing capacity and pockets. Make sure that it is durable and comfortable, and does not put additional pressure on your shoulders. Use it wisely by stocking up the heavier items at the bottom and lighter goods at the top.
Install a rear rack to your bike which is an easy way to increase the storage capacity. There are insulated panniers and bags available which allow you to store frozen items. You can further use various attachments which enable you to carry plenty of goods on your way back home.