Instead of choosing paper or plastic, you can purchase a tote bag (or find one around the house) to use as a bag to carry groceries and other things you buy when you are shopping. They are sold at Whole Foods for under $10 if you do not have one. When you shop at Whole Foods, you receive a $.10 discount for every bag you reuse (yes, I know that's not much considering the enormous prices there).
I got a tote bag recently and at first I had a hard time remembering to bring it with me when I went to the store. To solve that problem, I started keeping the bag in the car. Then if I go shopping on a whim, I've already got it with me.
An informative, helpful website is ReusableBags.com and you can also buy some innovative bags on the site.