If at all possible, try to get a place on or near the
Campo de Fiori.
It's ever-so-slightly off the main routes, has a market there every day which is nice, is just a fantastic place to spend your time, and is completely walking distance to Trevi, Pantheon, Spanish Steps. Colosseum and Forum are a little further. I did the walk, but I'm a walker, I walk everywhere.
Oh man, yeah... go to AirBnB.com and type in "Piazza Campo de Fiori, Rome" and check out the places that come up. I seriously cannot say enough good things about AirBnB, I've stayed in a dozen of them all over the world and have had great experiences every single time, both with the people renting their places, and the company itself.
This place OMG:
https://www.airbnb.com/rooms/2577377?s=jXdc