A number of years ago, when recycling programs were just getting started, there was a catchphrase: "Reduce, Reuse, Recycle." It didn't seem to catch on very well, and I haven't seen it at all for quite some time, but it really makes sense. The trick is to do things in that order. First, reduce the amount of stuff you consume. Then, reuse the stuff you already have rather than getting new. Finally, recycle what you can't reuse any more. Do all that, and you have a lot less garbage on the curb on Monday morning (or whatever day they come in your area).
With grocery bags, we actually sometimes get plastic and sometimes get paper, depending on which one we need more. However, we do try to reduce the number we get. When the people in the store do the bagging, they assume you can't carry any bag that weighs over 2 ounces. So I just do my own bagging -- I'm faster than most of them, and I put a lot more in each bag. Also, when we're just getting milk or a paper or something, we ask for no bag (even though they often assume you want one).
We use both paper and plastic bags for a variety of things around the house. Paper ones are handy, because you have to put paper recycling in them to put it out on the curb. We're lucky where we live because we can recycle all sorts of paper, not just newspaper, so we really have a need for the paper bags. Plastic ones get used for many of the purposes people have already mentioned. Plus, we keep a bunch of them wadded up in the back of my car and we do take them into the grocery store to reuse them. Our store even encourages this by discounting your total by about 4 cents for every bag you bring yourself. The plastic ones are a lot easier to do this with, because you can wad up a bunch of them into a very small space.
Finally, when the bags are pretty well worn out (or with the really flimsy ones that some stores give you that don't last long at all), another grocery store chain near us has containers to recycle the plastic bags at every store.
So we don't reduce our use of these things to zero, but I think we do pretty well, and it really doesn't take a whole lot of extra effort. Now if we could just convince everyone to give it a try....
Steven