Have a look at
Porta Portese annunci gratuiti - il primo portale di annunci per Roma e Lazio
It's in Italian but is very good, get an Italian speaker to help you. For 500, you will only be able to rent a room. If you want your own monolocale (studio apartment) you should budget more like 700.
You will need a car or at least a scooter.
Most people come and stay a couple weeks in a b&b while they are looking for accommodation.
Start learning Italian, you will need it!
Good luck.