I just got back from Rome this past Thursday and loved it:
Not sure where you are studying in Rome, but here are my tips:
(1) Via del Corso - runs through out much of the city and all the main site(s) and then connects with Via del Fori Imperiali. So, pretty much it will take you to: Piazza Venezia, Imperial Forum, & Colosseum. Also, do not forget to stop by the numerous ice cream or gelato places along the way. Some delicious combinations and amazing culinary tastes. Also, numerous small cafes and restaurants as well to enjoy some real authentic Italian food. Enjoy a real macchiato, cafe or cappuccino.
(2) The Pantheon and Piazza Navonna can easily be located as well since there are signs from Via del Corso to side streets to get there.
(3) Via del Babuino well take you to Piazza di Spagna where the Spanish Embassy is and the famous Spanish Steps, and if you continue along that main road you will find Piazza Barberini, with incredible fountains and a main plaza.
(4) The famous Fountain of Trevi - now considered a UNESCO site is being restored, you can still go an see it, it is currently encircled in a plexiglass as they restore it and clean it up, so no running water in the fountains at the moment. It is scheduled to be completed restored in a year (but in Italian time) I would see a year and a half or two, everything in Italy runs at a different pace.
All these are easily accessible via walking you really do not need a car and I discourage it, most of the roads are now for pedestrians, you do see some taxis, and buses along the way and also Carabinieri (Police) they are trying to curtail pollution in the city. You will notice this also when you walk to the Imperail Forum most traffic curtailed to save the Colosseum from carbon monoxide pollution.
Another way to see the city if you are tired of walking which I did is via Skegway Rome Tours, this was a lot of fun riding a skegway through the city.
A word of caution keep your money in a secure place and also ID, although I saw a lot of presence of Carabinieri (Police) there are still pick pockets dying to get at your Euros, etc. Therefore, when I walked in the city I left my passport in the hotel safe and just walked with some Euros/one credit card and never put my wallet behind my pockets, etc. I carried a hidden money wallet. Also, not sure if you speak another language but you get treated better if you do. I found if I spoke Spanish which I am native rather than English. I got better service when I tried a bit of Italian or if I spoke Spanish.
I did not get a chance to go to Florence, Venice or Milan and those will be my next cities next time to go.
And one last thing Italian men are incredible hot and handsome, enjoy