One thing you cannot miss in Rome is an afternoon walking along the Via Appia Antica. Built in the 3rd century BC, it was called the "Queen of Roads", for its length, age and almost perfectly straight path. It was created to unite Rome with Brindisi, the ancient capital of Apulia. You need thirteen or fourteen days to explore its 530km. length.
The Appian Way begins more or less at the Baths of Caracalla. Today, the best preserved area is beyond the wall, just past the Porta San Sebastiano, and gives you an idea of the suburbs of the ancient city.