Heading to Rome (Collesium area) & Venice (St. Marco Sq)
Rome and Venice are two of the most discussed cities in Italy on this board. I would so a search for earlier posts for recommendations that might be suitable for you. If you can be more specific on budget, type of food, area (other than the area you are staying in), you might get some good recommendations. "Not expensive, not cheap" can be different for different people. The kids' ages might also be a factor; what is appropriate for a 16 year old teenager might not be for a 5 year old infant.
Except for a few high end restaurants and those in hotels, just about every eating places in Venice is family own if that qualifies as mom and pops. In Venice, hole in the wall can be charming with decent but rarely good food.
There are a few places near you, in the Monti neighborhood (about an 8 minute walk)
Taverna dei Fori Imperiali, Via Madonna dei Monti. Family run, delicious food.
L'Asino d'Oro, Via Boschetto 73. Newly opened, sophisticated take on the classics. Modern but relaxed atmosphere.
Trattoria Monti (much discussed) not so far away.
All of these are good choices, not too fancy, not bare bones. All good food.
Via di San Vito,13a, Roma , IT
Via del Boschetto 73, Rome, Lazio 00184, IT
Taverna dei Fori Imperiali
Via Madonna dei Monti 9, Rome, Lombardia 00184, IT