Most people in Manhattan either have them delivered from various grocery stores or from FreshDirect.com...whose warehouse, oddly is in Queens, yet they don't deliver to Queens, where I happen to live...but that's another answer for another question.
Most of my friends stop off at the supermarket or neighborhood grocer on the way home to pick up what they need for that day or the next few days.
Some of them...the less wealthy, will organize a group trip to Costco across the river (again in Queens), rent a van and 4 or 5 people will shop there and van it back to Manhattan.
Most people in the rest of the boros (Queens, Bronx, Staten Island and Brooklyn) will do it the way you do...once a week, once every two weeks or, if you can REALLY plan ahead, the way my family did it, once a month for pantry and freezer items...picking up only fresh produce and dairy as needed weekly.