The first rule of being an efficient grocery shopper is checking what you need before you leave the house and it's a rule which at one time, I rarely kept. I would push my trolley around and seize on yellow sticker bargains with great glee. BOGOF grabbed me every time and I honestly thought I was a savvy shopper. It wasn't helped by having a huge kitchen and storeroom with two fridges and two freezers. Things were bought because they were "bargains" and that was my idea of being canny!
I have to admit that I'm a tad obsessive with spreadsheets, but my food inventories certainly pay off. I'm not good at crossing stuff off or adding items as I buy them, but every month or so I take everything out of my freezer, one drawer at a time, and list what I've got while making a note of anything which needs using very soon. Regular cleaning and sorting of the fridge helps to keep en eye on food stocks and avoid waste.
I don't menu plan but I do meal plan. What's the difference? Well, I batch cook and I make a list of everything I need to make several portions of a particular dish but I don't plan when they will be on the menu. I use both fresh and frozen vegetables but even when I buy fresh I often freeze some of the stuff I buy rather than risk it going to waste.