As many people are probably typing this at the same time: bread goes stale faster in the fridge. If you need to preserve bread for a while, freezing is the way to go.
My goal with the bread is to keep it from molding, which is does in a couple days sitting on the counter. Usually it isn't around long enough to freeze.
That may be true, but in my experience it takes longer to go stale in the refrigerator than it does for it to mold outside the refrigerator. So I store my bread in the fridge and everything is fine.
I've never stored my bread outside the bag in something like a breadbox though, so maybe that would make it last longer. I've never had my bread go stale in the fridge before I finished it, so I'm not too worried about changing my habits.