This is the first time for us to travel with baby (1 year), so my story about it:

Appartements:
1. Appartements in the quite Italian patio, so your baby will sleep well.
2. It's on the first (ground) floor, so you won't worry about tight lift or dangerous stairs.
3. There are many heaters and warm floor, so you definitely won't cold.
4. There is a separate bedroom, so you won't disturb a baby.
5. Owners are very responsive people, we asked for additional baby bed and baby chair and they found this. Marco even bought a new baby chair especially for us, so you can use it too :)

Location:
1. There is not a so busy district with wide and smooth sidewalks, that is good for a stroller.
2. There is 2 pharmacies and supermarket. (Better baby food choice you can find in pharmacy)
3. Close to nice garden with beautiful view to Coliseum, when you can walk with a stroller.
4. Close to metropolitan station with lift. (Not all station in Rome has the lifts :/)
5. By metro, you can easily reach huge park and zoo, or Vatican for example.
6. Close to Central train station.

Finally, if you made a decision to visit Rome with a baby I recommend you this apartment and perfect owners.