I shop monthly. I didn't realize this was so unusual. I also live about 1/2 hour from town, and I drive an SUV so I try to minimize trips to town...but I also have space in my car to load up when I do shop.
I do my best to keep a fully stocked pantry, so I don't run out of something unexpectedly. For produce, I use those wonderful green bags that keep produce fresher longer. When I get home from the store, I divide the produce up into those and focus on using the most perishable things up first. By the end of the month we are usually eating more frozen/canned veggies and fruit.
We have an extra fridge and freezer, but I think we could still accomplish it with one fridge/freezer if we were careful about how the space was used. If you have any options for wintertime cold storage (an unheated room or closet that stays cool?) you could do some really basic root-cellar type storage there.
I usually end up making a very quick stop into the store for a couple of things once or twice later in the month, but only if we happen to be in town anyway, and only if we feel we desperately cannot live without the thing we're going into the store for. I try very hard not to go back into the store after that first big trip, because every time I shop I end up spending a little more than I expected, and the best thing for me is to stay out of the store entirely.
When I do my big shop, I try to do it at a time when I can take less kids with me (on a weekend or evening when DH is home). If it's only happening once a month, it doesn't cut into our family time as much, and I get a morning away by myself or with only 1-2 kids (which is a kind of a break, LOL).
To shop monthly you'll need to meal plan in advance, so you buy what you need and use what you buy.
I have been doing it this way for a couple of years now and I can't imagine grocery shopping more often than that. It's such a long trip and such a pain to do with kids in tow. Once I've done my monthly shopping (and that includes stopping at 2-4 stores), I don't have to think about it again until the next month. It feels much easier to me.
Edited to add that I often buy a little more than I think we need of snacks, staples, etc if they aren't already in my pantry...so we don't find ourselves short of something at the end of the month that would send us back to the store. I'd rather buy one extra bag of corn chips now and see it applied to next month's meal plan, than end up one *?&^$# bag short.
Also in meal-planning I am careful to take into account special occasions like birthdays, potlucks, picnics we might attend, holidays we are hosting or attending, camping trips, etc, so that I am not stuck without the right food for the occasion. This is the biggest reason I might end up back at the store - something came up last-minute that I couldn't manage to use what was on hand for. But I try hard to avoid those situations. I don't know how much food storage space you have, but be creative. It's good to always have ingredients for a couple of fallback potluck-type meals if you find yourself regularly needing a potluck dish (or unexpectedly entertaining a houseful of kids) last-minute.