Several folks below (including the Roaming Roman) and an emailer or two have recommended the Catacomb of St. Priscilla as the catacomb of choice…

Callixtus and the other catacomb site along or near the Via Appia are famous, and justly so,  but are also overwhelmed by tourists. You will feel like, well, a tourist.

However, at the Catacombs of Priscilla, you will be led by one of the Benedictine sisters who serve as caretakers or by one of their able assistants, young ladies who work for them. More likely than not, the tour will consist only of your group and (maybe) two or three other people. You will see some of the most important ancient Christian iconography, including the oldest known image of the Blessed Virgin (dating from the end of the 2nd century).  You enter the catacombs from the quiet little courtyard of the sisters’  convent.  Bring a light jacket or sweater for comfort.