"Slow" fast food in Rome and Siena?

We will be staying in apartments in Rome and Siena, so I will have a kitchen. We are travelling with kids, so eating in the apartment for a lot of meals is going to be more practical than restaurants. However, we don't want to miss out on all Italy has to offer gastronomically. Any suggestions on great take-out? (Other than pizza and panini.) Fun things to try from the grocery store? I understand there are stores which sell fresh-made pasta and sauces for you to take home and cook - what are the shops called so we can watch for them? Can anyone recommend some in Siena or Rome? Thanks!