Talking about Rome, I would reccommend Spanish Steps, but it can be very expensive.
otherwise, a good compromise between location and price could be the area surrounding the Vatican: there you got undergrounds, restaurants, attractions, museums...
This is my personal opinion: other people might thik that Termini Station Area would be better
I'm going to "fly by the seat of my pants" when traveling through Italy, will it be hard to find a hotel (at least 3 Star quality) without making reservations? I don't know when and where I'm going to be.