My town recently banned plastic shopping bags. YEAH! I know some folks find it a bit inconvenient but in general when I'm shopping especially in the grocery store I feel it's accepted. Partly because it's now the law.
At checkout you have a choice. You can bring your own bag. You can purchase a #5 PP 100% recyclable bag for between 50 and 99 cents. You can purchase a brown paper bag for 5 cents. You can carry everything out in your hands or cart.
The best and lowest waste choice is to bring your own reusable and washable bags. Being able to wash your bags is important if you want your bags to last. Clean bags also make your cashier happy. That's why I recommend canvas or other types of cloth. I have a lightweight but super sturdy bag I keep in my purse. It takes up very little room, it's strong, easy to pop in the laundry and that way I always have a bag.
Most folks opt for the plastic bags available right at checkout. Plastic #5 or -Polypropylene (PP) is a thermoplastic polymer. It is strong, tough and can handle some moisture. This plastic is used for kitchenware and containers. While it might be tough I find it rips easily especially when you put in boxes with sharp edges. You can mend the bags with duct tape to make them last longer. While they can handle some moisture you cannot put them in the washing machine so it's important to be a bit more mindful about keeping them clean. Use a damp sponge or cloth to wipe up food spills. The problem with these #5 bags is how to actually recycle them. I'm afraid most of them end up as trash because our local centers can't handle this type of plastic. That's a real problem with these product, while strong they will not last as long or be as easy to clean as a cloth bag and then it can be very difficult to get recycled. The toothbrush company Preserve has a mail in program and more info is here-https://www.preserve.eco/pages/how-to-recycle
For me paper bags seem like a better choice because they are made from trees and can decompose. These bags are not waste free however especially if only used one time. They are bulky and tend to rip especially if wet. When dumped in a landfill they need oxygen to break down so the process can take years.
The best choice for both you and the planet is to invest in sturdy, well made, washable grocery bags that will last and last and last.