The Jewish neighborhood holds some great ruins: https://www.cityexperiences.com/blog/jewish-ghetto-rome-history/. I was absolutely floored by the Teatro di Marcellus: https://www.turismoroma.it/en/places/theatre-marcellus. It was the model for the Colosseum and was started during the Roman Republic, the predecessor to the Empire. In comparison, the Colosseum is a kid. There are other ruins in the Jewish quarter, including the Temple of Bellona and the Temple of Apollo.
While in the neighborhood, the synagogue has an interesting museum. And the temple space, which your ticket to the museum entitles you to see, is beautiful.
Also, the early Christian Catacombs of St. Callixtus are fascinating: https://www.catacombesancallisto.it/en/index.php. You'll see the tombs of nine popes within. It's a necropolis under the city.
Nearby is the Appian Way, Rome's famed road to Bari, Italy. Rome conquered the world via its roads. You'll walk the actual, 2,000-year-old road while walking past monuments and other antiquities: https://www.rometoolkit.com/whattodo/appian_way.htm. You could also rent a bicycle as there are a couple rental places near the start, but be careful: The ancient road isn't always flat. You could walk or bicycle to Aqueduct Park to see ancient Roman aqueducts: https://www.romesightseeing.net/aqueduct-park/.
Finally, a visit to Ostia Antica, the ancient port town of Rome, is breathtaking: https://www.ostia-antica.org/. It's about an 1 hour, 15 minutes west of Rome.
