Is there a way I can find out places where I can stay for three months while I study there? I'm trying to keep away from tourist traps.
We used Cross Pollinate to look for places to stay in Rome and many of them are apartments that rent by the day, week or month. Most of them rent by the day or week but if you'll contact the owners of Cross Pollinate, they will let you know which ones rent by the month. They are quick to respond and are very helpful. http://www.cross-pollinate.com/new_seek.asp?lang=1&s=1
As Bobby stated, contact www.cross-pollinate.com They are a bed and breakfast and apartment booking service owned and operated by an American family living in Rome. I have used them many times and they are great. Good luck and have fun!!!
Go to Craigslist and select Rome then browse their apartment listings. Be careful of low rates--a lot are for tourists and charge a few hundred euro per week. If you don't mind sharing an apartment, check out the rooms for rent as well.
You can also try WantedinRome.com