Nice rail-related trip:
Use Lazali small gauge line from station next to Termini. Make a stopover at Porta Maggiore to enjoy the tram mix there. Then get on the small gauge line again to Lodi. There you can hop on Metro Line C which is running fully automated trains and runs partly overground. I did this a few years ago.
The Jonio-Branch of Metro Line B is also worth a visit, because its new
. I used this a lot, because my hotel was near "S. Annese Annibaliano" stop of this branch.
The transportation authority ATAC has a rolling stock exhibition at Porta San Paulo station.
For other sights:
Colloseum, Forum Romanum, Bath of Caracalla, Palatine Hill are a must. For these I bought a
Roma Archeologia at the Colloseum. It was worth the money and also included the museum "Palazzo Massimo alle Terme" (located near Termini) which was awesome.
In terms of churches I would recommend:
St Peter Basilica (including climbing up the steps to enjoy the view from the dome), Basilica of Saint Paul Outside the Walls and Santo Stefano al Monte Celio (aka. Santo Stefano Rotondo - a round church).
To get around a bought a 7 day pass (Carta Integrata Settimanale - CIS) from a ATAC ticket machine. It is valid on buses, trams, metro and FL rail lines within the city limit.