With a few changes in your habits and the right equipment, it's easy to do your grocery shopping and other errands on foot, by bike, or using transit.
Good reusable grocery bags can certainly make grocery shopping easier. I also use a backpack to put heavier items in. Another lesson I learned early on is to only use a hand basket when grocery shopping -- that way you won't take more groceries than you can carry at once. It's no good to load up a rolling grocery cart and realize at the check-out that you won't be able to carry it all home. This is where the change in habit comes into play. If you go to the grocery store more often, you'll have fewer groceries to carry each trip. If you're the type to load up a full cart every two weeks, then this may take some getting used to. Of course, if you live within a close walk to the grocery store, then going more often isn't a problem. And if you live in walkable neighborhood, then the walk can actually be enjoyable.