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I've been doing all my groceries with a detachable carry frame (Racktime Boxit Large) on my rear rack for the last year and a half. Pic for reference, not my bike, I usually put in a fabric folding box I found elsewhere that just happens to fit perfectly (Meori Large, do not ever use in soaking rain or wash in your shower because it's cardboard reinforced, I made this mistake after spilling something on it) instead of the standard rattly plastic tub that comes with the carry frame.
Upsides: Very sturdy even for carrying 20kg/44lbs worth of soft drinks (rated for 25kg but you could probably stand a person in it). 30L folding box just about works out for a week's worth of groceries for one person, sometimes I need to throw a few items in a backpack.
Downsides: Requires a compatible Racktime rear rack (was shopping for a bike anyway and found a decent one that already had one). With heavy loads it becomes slightly top heavy (noticable during turns but not worryingly or dangerously so). With heavy loads it also becomes very rear heavy (dismounting to lift the front wheel over a tall curb can have the bike wanting to topple over backwards and on one occasion it almost fell over into a parked car because of this).
Alternatives: The internet offers a million ways to fix a milk crate to any rack. A lot of racks are also compatible with pannier bags that cost more and don't fit items as large as milk crates do but offer a lower center of gravity. I've also been thinking about switching to a trailer (only extra part on your bike is a hitch) but I'm not sure it's necessary for my needs or even all that practical to store and use.
Tips: Weight distribution with lots of groceries on one end of your bike kinda sucks. If you're not obsessed with keeping your bike lightweight I suggest you to put both a front rack and a rear rack on it and use any combination of milk crates and pannier bags to distribute the weight along the length of your bike.
