Congratulations on your new baby. Nikon D5000 or D90 are good choices. The Canon D-SLRs are also very good.
There are a lot of fun photos you can do with a baby. Here are a few photos of my son at 6 days old several months ago.
The key is to set up your equipment before posing the baby for photos. You want to be quick, before he/she cries or before he/she gets cold if taking nude photos.
The most important equipment, besides your camera and lens, is lighting. Natural lighting is free, a large north facing window can be a good source of light for portraits. A cloudy day outdoor or sunset can also provide pleasing softer light. Try to avoid using your built-in flash. At least use an external bounce flash with a diffuser. If you want better lighting, use 2 flashes off camera. Or you can use strobes or other external lights. The above photos are done with a pair of hot tungsten lights (to keep my infant warm, this is the only time I ever used a hot light). Usually I try to avoid hot tungsten lights because they can get very hot and uncomfortable, but it can work well with newborns who need to be kept warm. For older babies and kids/adults, use strobes (or flashes if your budget is limited). If your budget is limited, then do not buy a tungsten light because you won't use them much at all. My tungsten lights are now warming the pet turtle tank now for my older son. Most nonprofessional people will just use their external flashes.
Any entry level D-SLR will be adequate for baby photos. The D90 will be a better choice later when your baby grows up and can run or do some sports. For babies, the kit 18-55mm lens should be okay, use the 50mm end. I used the good old 50mm f/1.4 prime lens for the photos above on a full frame D-SLR. The 35mm lens on the Nikon D5000 should also be okay. As the baby grows up, you will need a mid range tele lens 70-200mm.
Choosing a background is also important. A simple backdrop can make the photo more professional looking for announcements. You don't have to spend a lot of money for backdrops. Your local Linen shop can be a cheap way to get some black, white or colored background. Solid colors are usually less distracting. If you want something more fancy, then there are digital background that can have variable costs. You can download some for free (but usually low resolution). You can also buy them from a professional or web vendor. You can make your own digital backdrop too, by taking another photo and merge with the portrait.
This is a backdrop that I make on my own from an old photo of Disneyland. The portrait of my older son is done with a green screen and a set of studio strobes (monolights):
If you want to use digital backdrop, then you will need to buy a bright green bedsheet (from Walmart or Target) or a green screen from a pro camera shop. Then you take the portrait with the green background. And use Photoshop or other software to merge the photos by using layers and masks. These sound complicated but really not difficult to do. I'm not a pro and have not done this before. I watched a Youtube video (this kind of technique is called chroma key) and tried it on my own. It takes about 15 min or more initially but it gets faster later. If you are inpatient like me, then there are chroma key software that can automate the process and do it in 2 minutes (but a good software will set you back several hundred dollars, almost the cost of your camera).