Hi
First, congrats to you on your newborn.
I'm a newbie DSLR user too. The reason why I started to take more interest in photography and/or DSLR, and eventually bought one, is also because of my baby daughter. But it took me a few months of googling and surfing around forums such as CS to arrive at the decision. And I do encourage you to read up a bit more about photography and lighting, about the functions of each of the cameras available, about lenses, about aperture and shutter speeds, as well as how to compose pictures before making the decision. After all it's not a small sum of money, and you want to make it worthwhile.
I'll leave the technical specs and recommendations to the experts and seniors around, but thought I'll share some 'newbie practical advice', having just been through this phase. If you think a DSLR will give you better pictures, that's only half right. Yes, equipment plays a part, but the cameraman is also important. 'Cameraman' includes not just the person behind the camera, but also all the theories on lighting, composition, etc. In the initial weeks of playing with my DSLR, most of my pix were not usable, and many were out of focus. They were worse than those taken with PnS. The first usable pix came at around 300-400 shutter count (I don't know what's the standard, but this is what I just went through). So you have to be mentally prepared for this phase of your DSLR life. If you give up too early, then money is wasted.
Therefore, I encourage you to rethink carefully what you really want. Do you want to (a) improve your photography skills, or (b) just want to pay for the technology to get nice pix? If you want (a), a DLSR will give you the opportunity and flexibility to explore more than what a PnS can. If you want (b), and think that a DSLR allows you to achieve it, then as I said above, you are only half right, because it will take some effort to get it right. You may be better off with a PnS.