Can you share your tips? I've got a family of 5 (my kids are about 3-4 years younger than yours) and I can't get under $1,000.
#1 choose what you think is reasonable for your food budget and get a prepaid debit card or setup a checking account JUST for groceries and transfer money into it each pay day. I transfer money to our grocery account weekly and my wife gets to keep any money that's left over at the end of the week (the first part made sure our groceries stayed on budget and grocery spend has decreased by about $150 a month since she got to keep the extra money).
#2 Go to the gym and exercise routinely, people that exercise routinely tend to eat healthier and start thinking about food as "this will take X minutes of hard exercise to burn off." I'm an ultra endurance athlete, burn 1400+ calories in an hour and I still pass on food frequently either because it isn't healthy or it isn't worth it. If you are the one making food decisions your family will inherit your healthier choices.
#3 Related to #2 eliminate unhealthy foods and snacks, people that buy things like chips, cookies, ice cream and soda tend to spend a lot more money on them than they think. We tell the kids they can have as much fruit / veggies as they want but we carefully limit the less healthy snacks. Science has shown we have essentially NO will power, the best way to improve diet is to prevent yourself from making bad decisions by not having the food in the first place.
#4 Don't go organic for everything, organic is actually often worse for the environment because it's less efficient and requires more land / resources, many "natural" pesticides aren't better for the environment. Go organic when it makes sense but be smart about it, you usually get most the bang by going for reduced ingredient foods. Fruits and veggies I'll go organic but not if it's more than 75% more.
#4 Don't run all around town but know your stores and what tends to have good prices, we have Winco, Fred Meyer, Albertsons, Walmart (yuck), Co-ops, Trader Joes, and Whole Foods. Winco is great for the necessities but Fred Meyer is great for organic, their "Simply Organic" brand is surprisingly affordable and their produce tends to be better.