Publix has a great new campaign called "Bring Your Bags." Basically, they're encouraging you to "go green" and bring your own canvas bags to the store. BTW, they sell these bags in the store. It's brilliant...if their customers bring their own bags, then Publix has to purchase less paper/plastic bags and it saves them money...and they get to sell you Publix branded canvas bags.
I've been bringing my bags to the store with me for about 3 years now. I used to forget so now I just keep them in my car. I think they're way superior to paper & plastic. Environmental aspects aside, the canvas bags perform better. You can fill them with canned goods without having to double bag or worry about the bag breaking and sending its contents straight to the ground. You can put more stuff in each bag which means fewer trips while unloading.
I guess "canvas" might be the wrong word to use. There are other materials (nylon) that reusable bags are made out of. The FAU Enviro Club was selling Chico Bags a year ago as a fundraiser. Chico Bags fit in your pocket/purse and open up when you need them. They fold into their own pocket, kinda like some of those lightweight camping shorts.
If you're not yet in the habit of taking your bags with you when you go to the store, I recommend that you make a conscious effort to try it out. Give it at least 5 shopping trips and see how you like it. If you hate bringing your own bags, then go back to paper & plastic.