My wife had our first son in April of 2008, and in December of 2007 I bought a Canon HG10 for around $900. It's a HD camcorder with a built in 40GB hard drive. I didn't want to get all sorts of other media, so that's why I got something with the built in storage. It took decent videos and I have nothing to complain about.
However, speaking from experience, I found it quite the hassle to grab to film something. Transferring videos to your computer is also more difficult than simply plugging in some sort of media into a reader. In the end, I didn't use it very often! So here it sits, collecting dust, as I try to sell it on eBay...
Instead, I use my point & shoot Canon SD1100 camera to do most of my filming. It's so much smaller, I find it much more convenient. Keep in mind, I only do short clips (30-60 seconds), so if you want to shoot for a few minutes at a time, then maybe you should get a camcorder after all.