Last month, when I got my very first ever sewing machine, the first thing I sewed was a couple of cloth produce bags. I use these little beauties whenever I get fruit and veg, rather than using plastic tear-off produce bags. Since having them, I think I've saved about 20 of those little plastic baggies from going into landfill. Imagine the impact over a life time! (And see how comfy my lemons look nestled in their cloth baggie).
I didn't use a pattern, just made up a basic draw string bag. Used some 8 ply cotton as the "string". The fabric for the bags was a sheet I bought at a thrift store for $3. Thrift store sheets are bargain fabric finds!
Betz White has a tutorial on her blog showing how to make a shoe bag, which is essentially the same thing, if you need a little inspiration.
They are so handy, I just need to make a few more... some smaller ones would be good too, as my current ones are about 25x40cm.
Don't be shy about using them either! The feedback I have got from cashiers has been overwhelmingly positive. I'm not sure how much the bags weigh, but I'm sure it only adds a couple of cents to my purchase, and I think it's well worth spending a few extra cents to keep a bit of plastic out of landfill. (Some cashiers do offer to take the fruit out of the bag for weighing, but I never bother).