I used to live there, if you're only there for a while you're probably stuck visiting all the tourist-like places.
The quietest place is to head out to the Via Appia, there are a few ruins and it's usually quite quiet a beautiful. The Pyramid of Gaius Cestius is nice too, as is the Protestant Cemetary beside, both are also somewhat quiet. The Capitoline Museums have a second site heading south (one stop after Pyramide) and it's almost always forgotten by the tourist crowd. In the main Capitoline Museum be sure to visit the Tabularium (it's up a stairwell off the underground passage between the two Museum buildings) because it has the best views of the forum and it's neglected.
They've recently changed the rules for visiting the forum - it used to be open and free and you just paid for the Palatine, but now the Palatine, Forum and Colloseum are all part of the same "site", so it's best to get to the Colloseum EARLY, because it gets busy and then use the tickets to see the Forum and Palatine, for the forum doesn't get busy until a bit later and the Palatine almost not at all.