Plastic bags don’t biodegrade, they photo-degrade, breaking down into small toxic bits contaminating soil and waterways and entering the food-chain when mistaken for zooplankton or jellyfish. Many large stores have bins for used plastic bags. They aren’t actually recycled, but they are disposed of as safely as possible.
You can minimize your use of disposable plastic bags. Use a Reusable bag for groceries and other purchases. If you only have a couple of items, carry them in your hands instead of a bag. Many stores sell reusable bags for a reasonable price, and some offer a small discount if you carry in your own bag. If you’re picking up take out food, don’t use a plastic bag if you don’t need it.
You can also do something like buy different colored bags from ChicoBag.com then use one color for groceries, one for clothes shopping, etc. Their bags are small enough to fit in your pocket.