For video camera, I'd get something economical like Flip HD (we have Kodak Zi8 ) so you won't feel annoyed about it if you find yourself not using it all that much. Seven and half weeks in, we've taken the video of our newborn twice - once with the Kodak camera and another time with my iPhone.
I second (or third, or fourth, etc) about having a camera that you'd carry with you. For me, it's been my Canon DSLR (I got into a habit of taking it virtually everywhere I go), but for my wife, it's a P&S.