I agree with Pedex. There are many variables. If you live in a large city with fairly cheap, accessable public transportation then you can ride a bike/walk/take public transport with minimal cost compared to owning a car. If you live in a smaller city that has little access to public transport, it becomes a little more difficult (though not impossible). At that point many would suggest intentionally seeking housing near common shopping locations (grocery and such). That will make it easier to do basic weekly shopping trips. Large purchases, unless you're very adventurous are often easier to purchase online/through stores that offer home delivery.

You really end up planning your outings more. If you know rain is possible for the next two days, then it is probably better to head out and take care of shopping today. Take it from someone caught in the rain with shopping bags, no fun walking several miles in the rain with paper shopping bags!

Overall though, you can easily save a great deal of money. I use maybe $50 every 2-3 months on public transit and next to nothing on maintaining my bike, and nothing on walking. Compared to my friends with $3-400/month car loan bills, $300-600/6 month car insurance bills, regular vehicle maintenance bills, and the $2.50-3.50/gallon gas prices I easily save a good $500/month by not having a car.