The more effort you put in, the more you will save
Riding my bike to the store also really helps, as it means being much more intentional with what I buy. I realize not every one is able to do that, but for those that can, it makes a difference.
Taking greater care on what to eat helps with this plan. If you eat less and more carefully, by design, you will shop for less and more carefully as a consequence. It is a lot of work but putting more effort in to start with gets you further along than making zero change to your habit.
If there are multiple grocery stores nearby, checking the weekly sales schedules and making a list for each store is helpful. My parents, children of the Great Depression, used this strategy. I do sometimes work around the weekly sales for meat, chicken and fish, and have created room in my home to stock up on nonperishables when I find good prices.
Definitely police your leftovers and adopt the FIFO approach, that is an easy tip to follow.
Good stuff. I cook most of our meals at home, and this also keeps us from going out or ordering from a restaurant to be delivered. That secondary benefit is really the main thing for us: if we only eat meals we don't prepare once or twice a week (instead of 5 or 10 times), we're going to save many hundreds each month.