American here, so I’m automobile-dependent, but I most often just put the individual items directly in my trunk, without bags. I do keep a collapsible bin in my trunk that I use when I get home to carry in a bunch of the smaller items. My wife usually tries to give me a reusable bag to use when i head out until I give her a half confused look. then she realizes there’s near-zero chance that I’ll remember to take the bag in the store.