Reddit mentions of Promaster SystemPRO Basic 2-Light Studio Reflector

You can buy LED lights if you aren't interested in strobes, but you never know.. maybe you'll really take off! Before I got my strobes, I used these from Promaster for lighting smaller areas and they worked like a charm (keep in mind they can get a little hot - don't cook the baby!). But if you're interested in something a little more professional:

The strobe

I personally use a single AlienBee B800. You may be able to photograph newborns with a B400 (a bit more affordable, but pumps out a little less light) version since it won't require you to light many large areas.

The modifier

Then you'll need a modifier to put on it so you can spread that light around the set cleanly. I prefer to go with Fotodiox products, as they are cheap and affective. I own the Fotodiox 36" Octabox.

Then of course, you'll need decent light stand to put them on.

Depending on how you position your strobe, you may also want to get a little reflector disc so that you can bounce light into areas that are too shadowy in your shot. You can also just do this with a regular ol' white foamcore board.