I've never lived in a huge city, but where I live (Estonia), city planning is much better than much of North America so usually you wouldn't live too far from a grocery store, walking is usually an option, otherwise there's public transit. There's literally nothing weird about bringing your grocery bags on a bus or train or whatever.

If you don't have a car, you just shop for one or two days at a time usually. This way you can actually get fresher ingredients sometimes and you don't have to plan an entire week ahead at once. In fact, I'm lazy and just shop for one or two days at a time despite having a car and sometimes using it to go to the grocery store. I just don't know what I'll want to eat 2 days from now!