The number one tip to save on groceries is to get it to where you make less trips to the grocery store.
Go as few times a month as you can. At least 1 time a week or once a month if possible.
Buy a freezer and fill it up gradually. When there are sales on items that you know you will use and can be frozen buy as much as you can on your budget and fill the freezer. Even if you just buy 1 item extra that will add up.
If you meal plan this makes it easier. Get out all of the recipes that you plan on cooking for the week and just buy what you need for them that week. Start cooking from scratch. Doing this is fun and usually have leftovers that you can then have later or the next day.
Make lists to take to the grocery store and stick to it.