I've rented apartments either with my husband or by myself in Lisbon, Paris, Aix-en-Provence, Florence, Rome, Venice, London, Bruges, Zandvoort, Amsterdam and Athens. I stay in apartments for comfort and to be on my own timetable. If I save a little money over staying in a hotel, that's a bonus.
The shortest stay I've ever had in an apartment is 5 nights. The longest is 8. As a single or couple, I'd never rent one for less than 4 nights. But for the 5 of you, it might be okay, keeping in mind that the shorter the stay, the more expensive the staples or food you buy might be if you can't take it with you on to your next lodging.
Every apartment I've rented had some kind of kitchen staples already in it. Those that I needed to buy, either for me or for us, certainly did not come close to obliterating any savings realized by sometimes eating at home.
I can't imagine that 4 adults and a teen could possibly eat out every lunch and dinner (assuming breakfast is provided by a hotel) more cheaply than preparing most breakfasts and some lunches and dinners at home.
Kitchen preparation and dining equipment will vary from place to place. I certainly wouldn't go grocery shopping before checking what's already in the apartment in the way of staples and what kind of cooking equipment there is. That will have a big influence on what you bring home.
You can get inspired by the combination of the cooking and serving equipment in the apartment and what you see while grocery shopping.
Part of the fun for me is grocery shopping in indoor supermarkets, in open air markets, in cheese shops, in delis and in bakeries. I like having a fridge where I can keep cold drinks, milk, eggs, meat, veggies, fruit, you name it.
One of my favorite things to do when staying in an apartment is to bring a prepared meal from the grocery store or takeaway from a restaurant home to heat in the microwave and supplement it with something I already have there.
I'm not a big foodie. Especially as a solo traveler, I tend to have breakfast at home, a nice lunch out and dinner back at home. That's my style and one way to save money on food. I like being able to have a simple sandwich from time to time and will sometimes bring one home even if home is a hotel room.
With 5 mouths to feed, I expect that your meal planning will be much more robust than mine is and that you're unlikely to have many leftovers.
Remember that in Italy you're never far away from great pizza and gelato, in France you're always close to mouth-watering croissants, baguettes and pastries and in Switzerland you'll be easily tempted by cheesy and chocolate fondues. Your inclinations and kitchen setups will determine where you choose to eat those.