I have fond memories of grocery shopping by bike. I've carried anything from live tropical fish to large packages of meat. I've picked up milk on the way home from work lots of times. I remember buying pop, back when the cheapest way to buy it was in eight-packs of glass bottles. Perhaps the easiest way to try practical cycling is by riding to the library, video store, or any other short trip not requiring a lot of cargo.
Backpacks also work, but they can get hot and heavy.
Some cyclists aren't comfortable in public places wearing bike clothes, but others have become accustomed to it. Any bicycle will do, but an efficient one will be more fun. If your bike needs work, fixing and modifying a bicycle is rewarding work. Bicycles are wonderfully simple to work on, requiring only a few special tools. Using fat, knobby tires on pavement will waste energy and make cycling less fun, but changing them to high pressure road tires is easy.