Going to the farmer's market means schlepping along your own stuff. It's busy, crowded and usually fast moving. With so much to see and purchase, make it easy on yourself and bring your own bags.
Crochet your own
From experience, reusing grocery bags can work, but they not very durable, and will more than likely tear just after they are filled with juicy tomatoes. If you have a stash of plastic bags, and a little talent with a hook, you can crochet a bag. Use the onion bag that is in your pantry right now, and some crochet stitches, for an easy peasy produce bag.
Knit your own
I love knitting. It just feels so earthy and motherly. Yes, I do need to get out more. Having said that, try knitting a produce bag or a larger tote for all your market goodies. A classic style string bag is amazingly expandable. Check out this one, using double pointed needles.
Buy your own
If you don't have the time or desire to make your own bags, don't despair! There are many talented crafters who will be more than happy to make some lovely, durable and unique produce/grocery bags for you.