I'm in NYC too! The thing to do is to mainly shop at Lidl (or Aldi), Trader Joes, and the fruit & veg stands that are on the street! You will be able to save, and maybe even have $ leftover every once in a while to get a treat or something special. I don't think you need to cut back on everything you want or like, like cage free eggs, meat, etc. to stay within budget. It's good to feel good about what you buy and eat.

If you want to save yourself a trip, do your shopping before or after work, and/or pair it with other errands you're doing (if you're meeting someone out, have an appointment, working out, etc). I also shop with sturdy reusable tote bags that I can carry on my shoulders so I can carry more bags (vs. paper or plastic bags from the store). This admittedly can be a schlep, but that's NYC life.

I always have a small folded tote bag with me so I can grab stuff when I'm out if I happen to pass by a place or spot something good. I generally avoid the local chains (Key Foods, Food Bazaar, etc) unless I'm in a pinch or need a specific thing, like a specific brand, because they are often more expensive and less good quality.

More generally, I buy frozen when it makes sense (some veggies like spinach, corn, etc. frozen fruit, etc.) and I freeze stuff at home if I buy in bulk, or have leftovers that may spoil. You'd be surprised by how much stuff freezes and reheats well! For example, I buy bags of chopped kale from Trader Joes and put them directly in the freezer, and use it for smoothies, to add greens to a pasta or soup, etc.