Reusable bags are the best choice, and, as with anything, it’s best if these are made from renewable, non-toxic and low-impact materials. Simply substituting disposable plastic bags with disposable paper bags is not a good solution.
Plastic bags are made from a non-renewable resource, petroleum, and they never fully break down. Even when they do break down, they simply become tiny flakes of plastic. In a landfill, it’s likely they’ll never break down as the landfill is compacted so tightly. While they can be recycled, contamination is a big problem, and recycling also takes energy and resources, and results in a lower quality plastic.
Paper bags are often made of recycled paper, and they are made from a renewable resource. They’re much easier to recycle (and people are more likely to recycle them according to many studies), and they break down quickly in the environment (if they were to get away).