It generally takes about 10 minutes for this to raise it's head when you tell someone you don't have a car. I've don't really understand why it seems to be a concern, but, then if my blind, non-driving mother could manage the shopping in rural England, suburban Australia is bound to seem a doddle.
First off, I'm only cooking for one these days (albeit, a disgustingly healthy appetite). I also like to pop to the shops a couple of times a week. I don't buy much frozen food, so, getting my veggies regularly keeps them fresh. Within 4kms I have 4 major chain supermarkets (yuck!), a wonderful Italian provisions store, an Asian grocers (lovely greens, sigh!), a grower's mart, a beef co-operative outlet (from down south), a delicatessen (over-priced), 2 small independent grocers (my usual), an organic food supplier, a middle-eastern supermarket (fab bulk grains and spices), 5 liquor-stores, a wet fish van, 2 butchers and 4 bakeries. I don't think my not having a car is liable to induce starvation any time soon! I'm not even counting fast food outlets, restaurants and the fact that, using the train, I could shop in any suburb I liked.
I tend to get my day to day supplies from IGA (an independent grocers 2kms along the road). If I were consciously thinking of doing a 'big' shop I'd pop the panniers on. The fact of the matter is though, that 90 percent of the time I just plonk my work bag back on the rack and head off. I lock the bike to a post that is less than 3m from the entrance and the check-out staff keep an eye on it for a little added security.
I think I've got an in-built mechanism for calculating how much will fit in my bag as I never seem to go over and rarely have much space to spare. I guess that might be the only difference in taking a bike to the shops. If I get what I need and there's still room I make purchases to fill the space with things that I may need later in the week, or that I can stock up on. Toilet paper is always a good one! This trolley easily packed into my work bag.
If I was shopping for two then I would probably use panniers more and if I were shopping for a family of four only once a week then I would use the trailer, but, it's still perfectly easy to do. If you're not used to packing a bike for camping there are a few things that it might help to consider. Packaging! Boxes, bottles and jars take up a lot more room than the same items soft-packed. So, for example, if I am buying spices I get the ones in bags and refill my glass jars at home and I buy my oats or muesli in bags rather than boxes. If I am buying laundry detergent or dish-washing liquid I get the concentrates. "It wouldn't fit on the bike", is a great reason for not purchasing plastic bottle, after plastic bottle of icky soft drink. Liquids are not only heavy, but, make the most disgusting mess if they leak.
I am not an advocate of disposable shopping bags, but, I always have a few and wrap my liquids in them. I don't want my groceries awash with OJ, honey or yoghurt. If you are using panniers, try and have them weighing roughly the same and pack the heaviest groceries at the bottom. If you want to maximise your room then don't just try and stuff your grocery bags into your panniers, use the little items to tuck into spaces. If your panniers or bag detach from the bike, remove them, pack them and then put them back. If you are packing a trailer try and keep the weight over the wheel base (I find the 2-wheeled trailers better for carting things around the 'burbs. They tend to be more stable and can carry greater amounts than the single-wheeled ones).
When I'm all packed it's then a gentle pootle down a couple of bike paths and within 5 minutes I'm home. By the time I'd found the car keys, combated the traffic, parked away from the store-front and walked, returned the trolley to the trolley-bay, started the car up, etc. I can't see how shopping by car would be any more stress-free or take less time.