I am a bike commuter and I am also a daily grocery shopper, which is a habit I picked up a number of years ago and haven't ever really been able to shake. It's not that I don't like to plan ahead- foresight is twice as good as two-sight!- it's just like I like to buy what I need only a meal or two ahead (aside from pantry staples) and this is especially important when you enjoy fresh produce and seemed to be cursed by a series of refrigerators that barely work and/or have been placed under some kind of evil curse to spoil food as quickly as possible. Or maybe I shop daily because I bike commute and not the other way around, but I do have access to a car and could theoretically just go once a week, pack the trunk to the brim with Funyuns and be done with it, but I don't do that, so the predilection for daily shopping must be deeper than just a habit forced upon me by my mode choice. Anyway. Maybe you don't like grocery shopping as much as I do (I think I have Stockholm Syndrome about it and that also explains why I'm always buying Swedish Fish) and maybe you like grocery shopping and prefer to do it by car and both of those things are ok with me, but if you're inclined to want to try to do your grocery shopping by bike or already do your grocery shopping by bike and want to compare methods (because you're a weirdly competitive person or something?), then I offer you the below.
Some thoughts about grocery shopping (and not Deng Xiaoping) by bike:
1. Have you ever driven a car in a grocery store parking lot? I've seen demolition derbies that are less barbarous. It's absurd. Everything you've ever wanted to learn about the malignancy of car culture can be learned in a Whole Foods parking lot on the day before a snow storm. It would make Thomas Hobbes blush (if he weren't so pale from being so dead). Rolling up to the store on your bicycle, for the most part, spares you from this malarkey. Bike parking, while not always great and not always sufficient, rarely requires the same selfish maneuvering or hastens the same kind of deep personal suffering as does circling in a crowded parking lot, trying to find a place for your car. And bike parking tends to be closer to the door, though if you're the kind of person who it put off by the idea of walking from the other end of the parking lot, you're probably not the kind of person who is going to bike there. In any case, choosing to bike to the store spares you a lot of the crazy car parking nonsense that so many people find to be so frustrating, so, as far as I'm concerned, you're already ahead. So would walking or taking the buses, but this is a bike blog, so woo bikes. Also, your bike can prove useful for carrying groceries, though a mule would also do that and you don't see me advocating shopping by mule. Yet.
2. I like to use my pannier as my grocery bag. For one, I'm already carrying it with me, so it's not an extra thing to remember. I walk around the store and put my groceries in there with my other stuff and when it gets full, I'm done. I've yet to have to make a Sophie's Choice, shoving my day's worn clothes behind the Entenmann's endcap in order to make more room for the the steeply discounted day-old muffins, but I suppose that's always an option. Maybe you're better than I am in judging spatial relationships, but I've definitely screwed up in estimating how much my bag could hold in relationship to how much I've already put in my basket (I don't use carts- they are cumbersome and unwieldy and should be left for people who are buying 11 dozen take-and-bake pizzas for some reason) and it's never a good thing when you're cramming stuff into your pannier that maybe you could've held off on buying. So...
3. Keep a spare bag (or two) on your bike or in your bike bag. The kind of bag that I really like to use in case of emergencies is a drawstring bag that you can wear as a backpack. I got mine from the AAA MidAtlantic table at last year's Bike to Work Day. [Previous sentence not intended to make your head explode]. What I like about it is that it folds (read: crumples) up very small for storage and can be worn. That way I don't have to sling it over my handlebars, which is a very "ugh" way to have to ride your bike. It can be done, but not by me, because I have very little balance and the bag ends up banging against the spokes of my front wheel and that makes both a terrible sound and perhaps puts me in a state of jeopardy vis-a-vis breaking my bike and falling down.
4. Don't wear your helmet in the store. Please don't do this. Please. I promise you that the risk of canned corn falling on your noggin' from the upper shelves is marginal at best. This is more an aesthetic appeal than a pragmatic one. I just think you look like a dork when you wear a bicycle helmet inside a grocery store.
5. Shop for what you need and be pretty rigorous about it. Going by bike probably ins't the best time to browse. Yes, you can probably (if you really work at it) figure out a way to bring home the 24 pack of toilet paper or the 36 pack of variety Utz snacks or a billion diapers, but if this is a daily shopping trip and you haven't already accounted for how you're going to easily do this, it might be better to wait for a different trip or maybe one in which you're not going by bike. Again, I'm not gonna tell you how to live your life. You do you. It might just be easier.
6. This isn't bike specific, but I really don't like using and then throwing out all of those plastic bags for produce, so I bought some reusable mesh bags. I guess I could've just kept reusing the same plastic bags and spared myself the additional expense, but I didn't do that and now I have these instead. I keep a few of them in my bike bag and try not to forget to put them back in my bike bag when they're empty again.
7. There are various bungee apparatuses that might prove useful to you, such as bungee cords and bungee nets and I have both of these things, but generally, I just try to fit everything I can into my bag and don't really avail myself of these things. But I have in the past, especially when I rode a bike with a front basket. I've used the bungee cords to secure growlers of beer and also pizza boxes, so their utility is unquestionable.
8. Remember to lock your bike. It would really suck to have it ridden away while you're inside.
9. You can definitely bike home eggs. Just put them near the top of your bag. Generally, the stratigraphy of my pannier looks like this: work lunch container (which is typically glass), work clothes, heavy or solid groceries (canned goods, milk, dried pasta, anything boxed, anything frozen, beer), then eggs, then produce. Try to construct even layers. I tend to buy a lot of kale, not because I even like kale, but because it serves as excellent padding for more delicate groceries. I also stuff kale in my shoes to keep my feet warm and as noise-dampening insulation in various home renovation projects. Never actually eaten the stuff, but gosh, it sure it useful. Bread is very tricky, because sandwich bread is squishy. Mostly I cope with this by not getting upset if my bread gets squished because, seriously, there are more important things in life. But I guess one way you could mitigate this by leaving the top of your bag open and leaving the bread near the top. Just not on rainy days.
10. About backpacks or messenger bags: you can definitely shop by bike with these, though they might not carry as much (though that's ok), especially if they're already full of your work clothes and/or laptop or whatever. That also means your groceries are resting up against your back and that might prove uncomfortable. There's maybe a decent amount of stuff you could cram in there, but then you also lose the ability to use a backup backpack, unless (and I've done this), you're willing to wear two backpacks to bike things home.
Everyone's grocery shopping needs are different, though for the most part, a bicycle is more than sufficient for your shopping and with just a tiny bit of preparation (especially if you're going to the store alone. You can, I bet, shop by bike with kids, and countless people do, but I've never done this so I can't speak to any additional steps you might want to take into consideration) and the simple recognition of the reality of your carrying capacity you can avoid the headaches of shopping by car and further find utility in the very useful tool that is your bike. If I can do it, anyone can do it.
I've doubtlessly forgot a ton of things. Leave your insights in the comments.
UPDATE: A thing I forgot: be very careful riding around grocery store parking lots. People, as I alluded to earlier, aren't always the best and more conscientious drivers in parking lots, so continue to be mindful of that and, you know, remain cognizant of the world around you, no matter how distracted you might be by your impending purchase of on-sale frozen burritos.
One other thing about grocery shopping by bike: maybe your preferred grocery store isn't very bike accessible and that's why you don't shop there by bike. I get that. I've written the above from the perspective of a very spoiled urban bike commuter who has access to ample grocery stores on his commute route and fully acknowledge that these conditions aren't universally shared. I wouldn't suggest that anyone taken upon themselves the decision to shop by bike in conditions where simply riding a bike are hostile, though I would, as always, suggest advocating your local policymakers and politicians for a change in those conditions, that you, if it is your inclination, can subsequently shop for your groceries by bicycle.
UPDATE 2: Do you live or shop in Arlington, VA? I once wrote this authoritative guide on bike parking at Arlington grocery stores and maybe it's still accurate!