Our grocery bill per month for 3 of us tends to be about $500. It's definitely lower in the summer, when I have all the veggies from the garden.
I do coupon quite a bit, and have found that if I"m careful about it, it does save me money. When I first started with them, I would end up buying junk we never needed, just because I had the coupon for it. Now, I've learned to only get it if we need it, will use it, and the coupon price is good. If I know there is a sale coming up, and there is a good coupon for that item, I'll buy coupons for it.
I don't find many produce or meat coupons, but they do exist

They're just not as common.
One thing that helps me save is the clearance grocery racks. Our store has a produce markdown rack, which is usually things that need to be used in the next 2 days or so. They tend to be marked down by about 50%. They also have a dairy clearance rack, where I"ve been getting a lot of milk recently. They have gallons of milk that expire in 2-3 days for $1/gallon. We go through quite a bit of milk, and if I don't think we're going to use it in time, then we make some pudding pops and put them in the freezer for later.
I've also been buying a lot more do-it-yourself snack stuff, and packing that for DH and myself, and pre-packing snack bags for DS. He loves the snack boxes of raisins, so I finally bought the big container of raisins and measure them out myself.
Definitely find if you have a bread outlet near you. We have a Nissen outlet nearby, and can always get bread, rolls, and such pretty cheap. IF you find a great deal on it, bread freezes pretty well too.
Good luck!!