Shame you only have 3 days as you will need 1 for St Peters and the Vatican museum alone (thats if you rush). Most of the sites are within 30-40 mins walking distance of the Vatican - Colosseum, Forum, Tevi Fountain, Spanish Steps, Piazza Navona ( lots more) so it shouldn't really matter where you stay. Old Rome is fairly flat so it is easy to walk and nice to get lost.
You can go for a traditional hotel or a family owned one (you actually stay at their residence) we did this which was cheaper (and grungier) which left us lots of money for great coffee (practically impossible to find a bad coffee) and good food.
Our Hotel was 2 streets from the Trevi Fountain which was a good base, close to the train station ( Barberini ). All depends on your budget. Have fun - wish i was going too!