My biggest advice is to start looking over blogs of people that feed their families on budget. One blog I have gotten great ideas is:
grocerycartchallenge.blogspot.com
She feeds her family of 6 (including teenagers!) for $60/ week. Look through her blog for ideas of meals that are inexpensive and good. I have followed some of the recipes and have been very pleased. There are also links to other people's menu plans and budgets.
I have never been able to get our budget to $60/ week but I have gotten it to $80/ week with a fair amount of organics. My strategy was:
1-Shop ONLY store sales.
2-Go to more than one store to shop sales.
3-Meat (chicken or turkey) 1X/ week. All the rest are vegetarian.
4-Know the cheapest prices for the things you buy-- I always know the cheapest price that I have paid and where. For example, organic milk at Costco is 3.33/ half-gallon, but one store near us regularly has it on sale for less than that-- so when it is on sale I will stock up and make yogurt or ricotta.
5-Waste NO food. Eat leftovers, give kids small portions so there is no waste.
6-Minimal processed food. I make pizza, I make our breakfasts (the only cereal I buy is millet puffs for my gluten-intolerant husband), I make bread, yogurt, salad dressing.
7-Buy in bulk what you will TRULY use. I buy flour, sugar and oatmeal in bulk. I do use these regularly and this does save money over time, although it is a higher initial cash out-lay.
8-For us, I studied what foods were important to buy organic and what was less so, and I followed that.
I think that is about it! Good luck! I do think you will be able to get your budget to $300!