I think just about any casual restaurant, which is most of them, is fine with kids, families eating out is a common sight. I used to say that if you see the terms Osteria or Trattoria to describe a restaurant, that indicates a good basic restaurant, not too formal, and moderately priced, however those terms are used pretty loosely these days, but still an indicator.
I hesitate to give recommendations, due to personal taste and since I have not been back to Rome in a couple years, things change fast. But one example of a historically good place is da Enzo, a trattoria in the Trastevere, often indicated as da Enzo al 29 (29 being the address). They serve solid Roman dishes, simple pastas, artichokes, all done well.
I like to do searches for blogs on Italian and Roman food, if I read the blog for a while, I get a sense of what they like and if it matches my tastes. You can get great recommendations for smaller places, just out of the tourist zones. Just search something like "Rome food blogs". Katie Parla is a long time, well known, blogger https://katieparla.com/city-guides/rome/ and good for a start. But there are dozens, including looking at some YouTube vlogs.
I also just do a lot of walking around, looking at places and the menu, after a bit you get a sense of what the place is like, I rarely eat at a place sight unseen, even if it is on my short list, I want to walk by.
I can't help you with Michelin Starred places, there are close to 20 of them. Most have a style of food I just do not resonate with, I am sure it is great, more "Italian inspired" than Italian. I am sure you can ignore whether a place is Michelin starred or not, look more to the menu, your likes, and budget. There are many classic fine dining restaurants serving great local food.
A Michelin starred place, you would want to reserve long ahead, a very good upper range restaurant, a month or more out. For most of the other places, you can walk up and get a table. At most, if I have my heart set on a place, I go in person and make a reservation the day before, or earlier in the day. Keep in mind, lunches often, for locals, are still a larger meal often, than dinner is, so rarely is a "lunch menu" a lesser thing.