I took DS to supermarket when he was just a few days' old, and to a restaurant when he was 1 week old. We always go to the supermarket on weekday mornings, so it's pretty empty anyway. As long as you don't let people get close to your LO, you'll be fine.
FWIW, it was SO MUCH easier to take DS to a restaurant when he was a brand newborn, cuz he just slept through it. Now at 6wks, it is actually much harder to take him out to eat cuz he fusses a lot! We tried to go out for lunch last weekend but we never made it through the restaurant's front door b/c he wouldn't stop crying. Also, I'm not comfortable to NIP yet, so any outings are limited to a 2-hr window for now.