The Aligre quartier is so full of good stuff that recommendations are almost unnecessary. Rue d’Aligre market street, the morning street market (daily except Mon) on place d’Aligre, the excellent covered market Marché Beauvau (open Tue to Sat AM &PM with some vendors taking a long lunch break and Sun AM) make it one of, if not “the” food-fab quartier of Paris. Its vibrancy and unmanicured Parisness may be too much for some older tourists not used to city life but, for others the “hum” and abundance of choices can be intoxicating.
For takeaway, the covered market Marché Beauvau on the place d’Aligre is a treasure trove. At least two excellent rôtisseries (roast chicken at one and pork/ lamb at the other), cheesemongers (including the excellent Fromagerie Hardouin), a few charcuterie stands that also include prepared side dishes, poissonneries for platters of shucked oysters to take back to your apartment, an artisanal beer shop, coffee roaster, etc etc.
I also enjoy browsing the Aladdin’s cave of Mediterranean goodies like lokum, nougat, dates, olives, spices, nuts, at the Lebanese épicerie Sabah on the rue Faubourg St Antoine just next to Boulangerie Ounissi.
Since there are so many fishmongers, butchers, charcuteries, etc clustered on the rue d’Aligre and in the Marché Beauvau, it’s also a great opportunity for exploring the house-made rillettes, soups, etc that such shops sometimes offer.
Boulangeries/ pâtisseries. I doubt that any other quartier in Paris can match the number of choices in the Aligre quartier/ Faubourg St Antoine. Maybe 7 or 8 good-to-excellent ones and 4 or 5 more run-of-the-mill in a 10-minute walking radius. Since it’s so close to your place, Ounissi on rue Faubourg St Antoine is more than good enough for your daily croissants and baguettes. I’d also try Farine & O and Moulin à Pain, both almost next door to each other on the rue Faubourg St Antoine for comparing/ contrasting and finding the breakfast pastries that most match your own tastes. I find Moisan suggested by @sfcarole much less good than it used to be but still worth a try. Since you are staying in the area for so long, I’d also check out the highly regarded Blé Sucré on rue Antoine Vollon across from Square Trousseau as well as my own particular fave Boulangerie Bo on rue Charenton/ rue Castelar… maybe not a place for conventional trad pastries but a great place to experiment with modern variations. I haven’t tried a large range of his pastries but the lemon tarte at Cyril Lignac Pâtisserie on rue Paul Bert is, for me addictive and my instincts tell me his other pastries are probably also very good. Also starting to have a fondness for the new-ish pâtisserie (name?) inside the covered Marché Beauvau because of its strange mix of Mexican and French pastries/ cookies and charming French-Mexicaine couple that run the place. Given the length of your stay, I’d also sample the Algerian/ North African pastries at La Bague de Kenza on rue Faubourg St Antoine and/or Amira on rue d’Aligre.
Cavistes/ Wine bars. I’m not a wine geek and always a drink a wine with food, love pairing the food and wine, and, at a restaurant, almost always order the wine suggested by the waiter/ somm. For grabbing a few bottles for taking back to your apartment, the branch of the Nicolas chain on rue Crozatier is very convenient and the selections are good enough for us mortals. In my one and only experience there, the young guy helping me was exceptionally helpful, knowledgeable… and, mercifully, no up-selling or attitude. If Nicolas isn’t rarified enough for you, there must be a dozen cavistes and wine-bars in a 10-minute walking radius. I used to love Caves à Prague on rue Prague for the food more than the wine but, as @Carmenere has pointed out, the food is far less of a draw now that the original food team has gone elsewhere. Time-warp Le Baron Rouge wine-bar on the corner of rue de Cotte and place d’Aligre (side street) can be very enjoyable, especially on weekends Oct to April when they also serve oysters, but a little warning: it’s sometimes overwhelmed by organized tour groups attracted by its cutesiness and olde-worlde setting.
Crêperies. The Aligre quartier is not exactly good crêpe territory but a few very worthy crêperies in easy walking distance. If you can tolerate the cramped space, tiny Crêperie Le Goéland on place d’Aligre is very good… in addition to traditional French crepes, they also do excellent Lebanese-style rolled crêpes. There is also a branch of the excellent, very tourist-pleasing and somewhat upmarket Breizh Café crêperie chain on rue Paul Bert (less than a 700-metre walk from your apartment). I haven’t been to either in years but a similar walk in the other direction towards the Gare de Lyon, Au Beurre Salé on avenue Ledru-Rollin and Les Embruns on rue de Lyon are maybe even better as family crêperies because of their very good price/ quality ratios and space.
Restaurants. For me, Virtus on rue de Cotte is the belle of the ball. Admittedly the food is less good than pre-Covid when it was cheffed by a charming Asian-Argentinian duo but still worthy of its Michelin star. And a very relaxed and joyful ambiance (something somewhat rare for Michelin-starred restaurants). Yet, considering the Michelin star and the prices, maybe a bit of a waste with kids in tow. Jouvence on rue Faubourg St Antoine is star-worthy and has a startlingly good price/quality ratio for both lunch and dinner that makes it perhaps a better “foodie” choice for families… and A+ for ambiance and service… and it has onzieme’s seal of approval. What more could you ask for ?
For noshing en famille, I’d also suggest:
the carnivore’s delight butcher+resto combo Les Provinces on rue d’Aligre;
the pescatarian’s delight SeaBar on rue Crozatier/ rue d’Aligre;
the fishy Bar à Iode on rue Faidherbe at rue Jean-Macé, good value for family dining, and an opportunity for a before and after— or at least a photoshoot-- across the street at the ultra-picturesque Le Pure Café often used as a film location (but don’t ask me which films);
L’Ebauchoir on rue Cîteaux has such a good price/ quality ratio for the food and a very pleasing ambiance that I’d still recommend it as a good choice for families despite onz’s criticism of the wine;
Le Chat Bossu brasserie, also one of the neighbourhood’s prime hangouts but more one-size-fits-all than foodie-fab) on rue de Cotte/ rue Faubourg St Antoine;
Le Bistro du Peintre on rue Ledru Rollin at rue Charonne for “correct” trad, good prices, a large menu, continuous hours, Art-Nouveau décor and history, mixed clientele including tourists (they’re the ones eating soupe à l’oignon gratinée) and relatively good restaurant theatre likely to appeal to the kids as long as you insist on a table in the main dining room);
Le Chalet Savoyard on rue Charonne for relatively good value fondue, raclette, tartiflette, and other Alpine winter specialties… I wouldn’t be caught dead here in summer but do enjoy it in winter to remind me of my last skiing trip… and kids love this type of cheesy food.
Bouillons. I used to be a fan of the Petit Bouillon Pharamond after its relaunch a few years ago because of the value, décor, and ambiance but last meal there a few months ago was a disappointment. I imagine the chef(s) have become a bit bored doing the same trad dishes day after day, and standards have slipped. And it’s a no-reservation place and always a wait… I suppose you could slip in for an early meal at, say, 6:30pm without much of a wait but the trade-off is an empty restaurant which for some (i.e. me) is pretty joyless and un-Parisian. I much prefer Bouillon République now that they offer reservations but less enthusiastic about it than the new “old” Brasserie Martin near the Square Gardette (a 20-minute walk for you guys or walk down to rue Reuilly to catch the #46 bus to the St Ambroise stop). Brasserie Martin and its older siblings Brasserie Bellanger in the 10th and Brasserie Dubillot in the 2nd seem to me to capture the brasserie spirit-- and a less old fogey one-- far better than the more famous bouillons and landmark brasseries… and food is “fait-maison”, authentic, no-suprises brasserie fare at very fair prices.
The Faubourg St Antoine straddling the 11th and 12th is riddled with 18th and 19th century passages and hidden courtyards left over from the time when this was the furniture-making district. Wonderful to explore as long as you do your research and know where to find each particular passage and “cours”. Many are difficult to find. And not all are as interesting as others.