Congratulation! It takes time to learn how to manually adjust the D-SLR to optimize your photo, but it is really not difficult to use at all.
For a beginner, you should buy an entry level D-SLR. What kind of zoom lens you want is up to your budget. The kit lens is cheap but you will need to use flash indoor. A fast zoom lens will easily double the price of owning a D-SLR. My camera body is less than 1/3 the cost of my D-SLR system.
For a minimum beginner setup, you should have:
An entry level D-SLR with kit lens 18-55mm IS/VR
A telephoto lens 55-200mm or 70-200/300mm IS/VR will be a nice addition
An external flash that can tilt and swivel for bounce flash
A flash diffuser (Gary Fong Lightsphere is very good, but at least get a Sto-fen)
A camera bag
Lens cleaning kit
Memory card and extra battery are useful
If you have extra money left, a 50mm f/1.4 lens is quite good for baby photos, and allow you to take indoor photos without flash.
In the beginning, you will be mostly using Auto or preprogrammed scene modes (like how you use a PS camera). But for indoor photos, use the external flash with a diffuser, and you will see a significant improvement over the PS photos just by doing this. Learn how to bounce your flash to get the optimal result (and minimize unwanted shadows).
When you get enough cute pictures with the Auto modes, you can start using the Program mode with flash, then try the semi-Auto modes (aperture priority is mainly what you will use for baby and family portraits). The Program mode mainly let you adjust the ISO to expose the indoor or low light background, so that you can get a properly exposed subject and background. The semi-Auto modes lets you adjust aperture (aperture priority) or shutter speed (shutter priority) in addition to the ISO. Learn the aperture priority mode first, this helps you to learn how to adjust the depth of field and learn how to do selective focus.
If you have purchased a fast lens with a large aperture (eg. 50mm f/1.4 or 17-55mm f/2.8), then take off the external flash, and start experimenting how to take photos with a large aperture using ambient light without a flash.
Photography can be expensive, but you don't need to spend a lot of money to enjoy it or get great photos. You do need a lot of time to fine tune your knowledge and skill, but you dont' need to learn everything before your baby is born. As you learn more, you will start using more and more manual functions. It takes a lifetime to learn all the photographic tricks. I'm learning every time I use my camera, mostly from mistakes I made. The beauty of D-SLR is that you can review the photos right away and correct the mistake on the spot, so that you won't miss the photo you want to take. So don't worry about making mistakes, I have made pretty much every photographic mistake in the book. But the more mistakes you make, the more you learn and the better you get. However, you do need to learn how to correct the mistakes so that you won't keep making them in the future.