I think you'll enjoy Osteria di Monteverde and Tre Zucche.Both are chef driven, creative but in a relaxed trattoria type atmosphere. Unfortunately neither is very near your apartment and are in fact kind of far out of the center. But both are excellent and well worth the taxi ride. I also love Necci and Primo, both out in Pigneto. Again, a taxi ride away.
Closer to home is Romeo, the newly opened restaurant in Prati, a collaboration between Cristina Bowerman of Glass and the Roscioli brothers. Also in Prati is Sorpasso, which is a very relaxed atmosphere and delicious, simple food.
Urbana 47 is in Monti, and features organic local ingredients prepared creatively. Cool sort of industrial chic interiors.