I often stop off at the shops on my evening fitness run, and make one big shop a week. I have a light touring bike, with a cheap, but good Blackburn look-a-like rack, and 2 very tough Carradice pannier bags. I can carry more than I can pedal, up some of the steep slopes. An elastic bungi chord is also useful for bulky items on the top of the rack (6-pack of toilet roll). You can t break a bike with a shopping load.
A few times a year I rent a car or use a pool car from work and take the opertunity to stock up on bulky items and refil my wine and beer cellar.
On a bike I pay close attention to the altitude of the shop, esp when picking up heavier items.