What’s a great family friendly area to find low budget yet authentic food? Preferably accessible via public transit. Have rented an apartment near the Vatican area but we’re open to commuting a bit for the right price and flavour!
Head to the Trastevere area. Lots of great family friendly dining. Very quaint streets and out of the tourist frey.
You might get some ideas from this recent thread Rome Restaurants. I found most of the restaurants in Trastevere to be less interesting and desirable. There are too many groups of young Americans wandering the streets and the restaurants generally don't have to be good to be busy.
In Prati, the neighborhood near the Vatican, we really liked SiciliainBocca. Excellent Sicilian cuisine, fair prices, great service. Reserving is a good idea, but if you go early, i.e., at opening, you can probably get in without one.
La Pratolina is one of our top three pizza faves in Rome and is also in Prati, but do reserve in advance or be disappointed.