Plastic bags and paper bags are both bad for the environment. 
Plastic is not a renewable resource. And the EPA estimates that less than 5% of plastic bags get recycled. The rest of them end up in landfills, in forests, in the ocean, and a whole bunch of other places where we do not want them to be. Many animals eat them accidentally (especially in the water where the bags get mistaken for jelly fish). This can lead to choking, suffocation, or even starvation if the bag causes an intestinal blockage. It’s estimated that plastic bags kill about 100,000 animals a year that way. And they take a very long time to break down, hundreds of years at least.
Paper isn’t much better. Paper is a renewable resource. But as is, we’re cutting down trees way faster than we’re planting them. And paper bags are recyclable, but the EPA estimates that only 20% of them get recycled. Most of the rest end up in a landfill. Now that might not sound so bad because paper bags are biodegradable. Biodegradable things are things that can decompose naturally (we usually say that something is biodegradable when it also breaks down fairly quickly, but that isn’t technically part of the definition). So things like apple cores, banana peels, and egg shells. You’d think that paper bags and all that other stuff would break down no problem in a landfill. But they don’t. Because in order to break down, especially in a relatively short amount of time, things need air, and preferably some dirt and microorganisms too. And that stuff is lacking in landfills. Landfills don’t have much dirt. They don’t have many microorganisms. They don’t even have much air because everything in a landfill is packed so tight that it can hardly breathe. So stuff that could break down in a compost pile in just a couple of weeks may take years or even decades to break down in a landfill. So all those paper bags end up just sitting there.
So don’t leave the grocery store with paper or plastic bags. Bring your own reusable bags to the store instead. Reusable bags are much better for the environment, they’re stronger, they can carry more without ripping, and they’re easier to carry (have you ever tried to carry a full paper bag and ended up having to cradle it in your arms like a baby or tried to carry a full plastic bag and had the thin handles cut into your fingers!?) If you don’t have any reusable bags, you can buy them at most stores for a dollar or two. You can even make your own out of old shirts, even if you don’t know how to sew.
Sources:
http://www.care2.com/greenliving/paper-or-plastic.html
http://environment.about.com/od/recycling/a/biodegradable.htm