In baby photography one of the biggest challenges is lighting.
Lighting babies and children is something that comes with experience. As with lighting any other subject, the most important thing is that you become familiar with a few basic setups, and then adjust the setup to suit the particular situation and subject.
When it comes to diffusing the light you have a choice of brollies or softboxes. My preference, in the studio is the softbox. When I rolled out customers home I used translucent shoot-through brollies. One of the main advantages with the softbox is that it leads all of the light forward. With the shoot-through brolly some of the light escapes from the edges and bounces back toward the camera. With this you need to be careful of flare so always shoot some test shots first and remember to use a lens hood.
Translucent brollies also give a nice effect if you shoot the flashgun back into them and reflect it back out toward the subject. To keep the lighting soft you do need to get in very close if you use this method. The reason for this is that you increase the distance of the light from flash to subject, and as you know a distant light causes a harsher shadow.
Reflective brollies are less useful than the translucent shoot-through style; they are too harsh for most types of portrait work. That said if you are looking for a hard light they are fine, but I can not imagine a parent being happy with a harshly lit shot of their baby! Reflective brollies are however good for lighting backgrounds.
The main reason for keeping lighting simple for baby photography is their volatility! For example you could be ready to take the shot and the baby crawls out of 'position' … if your lights are 'clinically positioned' and very directional the baby will quickly move out of the shooting zone.
If you keep your lighting basic, for example two large soft boxes, then it gives you more latitude for movement.
A relatively flat, even light generally gives the most pleasing results for baby photography. As a starting point I like to place a main light, either reflected from an umbrella or diffused through a softbox, between 35 and 45 degrees to one side of the subject, then use a similarly diffused fill light at 1/3 to less exposure next to the camera as a fill light. I then adjust the lights to suit the subject's features in a flattering way.
An extremely fine-featured child often benefits by making the light "harder".
To achieve this, move the main light closer to the 45 degree position. Conversely, a child with large, sharp features or skin blemishes, such as a rash, will benefit if you move the light closer to the 35 degree position (ie back round closer to the camera) and lessen the power difference between the main and fill lights for a softer look.
When using a harder, more dramatic light on children, it is important not to over-do it. When seeking a more formal, dramatic effect, I usually shift the main light to between 45 and 55 degrees. The fill light should remain near the camera. You may want to lower the power of the fill light to deepen the shadow areas, but with baby photography I generally find that it's best to keep the ratio within one stop.