For light loads there's nothing wrong with using a bag, just as you would walking. As long as the load is stable, doesn't interfere with your handling of the bike and isn't too heavy using a bag is a convenient option for many.
A bicycle however is an amazing device for greatly increasing your load carrying ablity, in fact there are some dedicated souls who have moved home entirely by bicycle.
A bicycle rack is a device which allows you to load up your bike. It bolts over the rear wheel and attaches to the seat stays and had a flat top. If you want to use your bike to cart stuff around a quality rack is a good start.
If you've got a rack you can put a basket on it. It doesn't need to be anything fancy, an old milk crate secrly attached to the rack makes a very useful container for plopping your bags of shopping in.
To be truly secure such a fixture needs to semi-permament, the last thing you want is to be coasting around a corner only to have the orange juice spilling down the drain. If the main use of your bicycle is shopping then there's nothing wrong with that, however if you want to use your bike for other purposes having such an arrangement could be inconenient.
Panniers are bags which attach to your bike, almost exclusively to a rack for rear panniers or special mounts over the front wheel for front panniers. They invariably employ some means of quick attachment to the rack making them convenient for stop / go shopping.
Panniers vary widely in quality and price. I'm afraid it's the old story of you get what you pay for, however spending good money on good panniers is a worthwhile investment. I've been using Orteleibs's back roller plus model for over three years and they're every bit as serviceable as the day I bought them. Hey, if they're good enough for a couple touring the world over seven years, they're good enough for me! Down the Road
Several panniers come with a means of carrying them like normal shopping bags. This is invaluable as it allows you to load your shopping directly in to your panniers, clip them on your bike and off you go!
If you're still in doubt about the ability of a bicycle to carry a serious amount of load then look no further than here: Moving a house by bicycle
Yeah, believe it or not you can get a trailer for your bicycle. There are several different types and they attach to your bike on the seat post or on to a special skewer (that's the bit that goes through your wheel).
Generally they come in one wheel or two wheel configurations. One wheel models have the advantage of better handling while on the bike and the two wheel models tend to be easier to handle off the bike but do compromise cornering a bit.