You are better off staying near Termini station or Trevi Fountain.
Everything is around this. You need to book at least 3-4 months in advance if you are a budget traveller. If you are late in the game, then look at places far away but near to a train station.
Travel within Rome
There is a train and bus pass which you can buy from the train or bus stations and which will allow you to use these bus and train unlimited number of times till it is valid
Always validate the tickets. Do not forget to do that before you board.