The port itself is an industrial area and nothing to look at.
If you want to go into Rome on your own is very easy. Walk or ride the free shuttle to the dock exit. Exit the small gate and purchase a BIRG ticket for the train. It's 9 euros, good all day for the train to and from Rome and the metro/bus in Rome. There's a news stand just outside the gate selling tickets and Rome maps. It's worth printing out a detailed map of St. Peters etc as the station is a few blocks from the Vatican. The train stops along the way at several stations including St. Peters and Termini. Check the schedules for your dates online before you go. Note holiday schedules are reduced especially at rush hour, when you're needing to get back to the boat.