Congrats on the upcoming baby!
Mine first one is coming in May and I got a 5D back in November to prepare. I had a Xti for four years so I know how to control a DSLR pretty well. That said, the 5D is very easy for taking pictures. You can set it to P mode and auto-iso and take very good photos. Obviously, the more work you and your wife put into it, the better the photos will be.
In terms of video, going DSLR is a poor choice unless you are committed to it. Video takes large amounts of memory, obvious lack of auto focus, and a need for processing/editing on the computer. This makes it hard to upload and share. (The video from the 5D looks SO good though, I can't wait to use it on the baby)
My advice is to get a video camera AND a photo camera. This will allow the wife to take video and you to take photos. I would go for a DSLR (you only get one chance to take the best possible photos). For a video camera, people don't care too much about quality, I have a panasonic HDC that I will use to quickly share and it was pretty cheap.