I have grocery panniers and they hold quite a bit, but they are a little expensive. Mine are similar to these:
Amazon.com: Banjo Brothers Grocery Pannier: Sports & Outdoors
They simply clip onto the bike rack so you can easily take them into the store and fill them, no need for grocery bags. i've had mine for about 4 years and they are as good as new.
I saw this way to make your own that looks pretty easy to do. I don't know how well they hold up though.
How to make Grocery Panniers for $4
Life is death. We all take turns. It's sacred to eat during our turn and be eaten when our turn is over. RichMahogany.