I have a pattern for a knitted bag made from plastic shopping bags. You cut the bags into strips, tie the strips together, and then knit it up into a bag. It's pretty sturdy...
But yeah, to answer your question, cloth bags are the way to go. Recycling plastic and paper are wonderful ideas and things to do, but they still use energy to complete the recycling process. Cloth doesn't.
Now if I could just persuade the baggers at the commissary that I'm really not insane when I hand them my knitted plastic bags....