First and foremost: don't use flash. This really startles the babies and often makes them cry. Trust me - I have four kids and made this mistake myself.
You should probably use focal lengths of 50mm or more - otherwise the proportions might get distorted too much. Usually the lenses work best when you close the aperture one or two steps.
If you don't want to constantly switch lenses, I would use the zoom, because it is more versatile.
While not really a lens topic: try to not only take photos down to the baby, but get on eye level with the child. Also, the parents will love pictures from them with the baby. If you take these shots make sure to close the aperture so that the baby and the mother/father are in focus.
Also a parents finger inside the baby's fist is a nice (if somewhat cheesy) motive.
Be aware that small babies change very fast. If you miss a shot you can't really recreate the situation weeks later. So plan before you shoot what shots you are going to take. And be patient.
And BTW: I don't think you need to upgrade your camera. It's not the camera that makes great photos it is you (and vice versa ;-) )
All the best for your shot.