While we are very far from car-free, my wife and I have committed (recently, this year) to running as many errands as possible without a car. Grocery shopping is major one, it is a regularly-ocurring vital errand. Grocery stores happen to be spaced strategically, so in any somewhat dense city, you are always within a mile or two of a grocery store, so its easy to get started.
Somethings we found out....
1) Its easier to deal with loads that are mounted in front of the bike rather than on a rear-rack.
2) Outdoor farmer's markets are a great way to get fresh (and often organic) local produce.
3) Instead of one marathon grocery shopping session, its necessary to go several times a week: Sat-morning farmer's market, Sunday-morning farmer's market, Mid-week safeway trip, emergency recipe ingredient runs (that's for me).
4) The safeway parking lot can be a dangerous place since motorists are not used to bikes.
Recently, my wife brought back one of the most useful pieces of conference swag ever: A thermally insulated backpack. For milk, eggs, and fish, it is indispensible. I put some "blue ice" in the compartment and wear it on my back to the store. After check out, I put the perishables in the compartment and am good to go.