If you want zero shutter lag so you don't miss anything, you want an SLR. I love posting this answer, because it is so right for so many people. My wife got a D50 specifically so she could take pictures of kids and has never regreted it. The D40 would be just as good for this purpose. They even have a "child photo" mode that selects the best settings for a typical child photo if you don't know much about these things.
The Nikon D40 is a great little camera, very easy to use and quite reasonably priced. It has a few "consumer friendly" totally automatic modes that make it very easy to use, but still offers total photographic control when you are ready to take charge. It will get you in the Nikon family which is a great place to be. If you buy accessories and lenses, you will be able to use everything on any Nikon that you might upgrade to later on.
Check out Nikon's "Picturetown" promotion, where they handed out 200 D40's in Georgetown, SC. http://www.stunningnikon.com/picturetown/
Here are a few reviews, in case you have not read them yet. Be sure to note that they are several pages long and some of the reviews also have some sample images that you can look at.
http://www.dpreview.com/reviews/nikond40/
http://www.steves-digicams.com/2007_reviews/nikon_d40.html
http://www.popphoto.com/cameras/3756/camera-test-nikon-d40.html
http://www.kenrockwell.com/nikon/d40/d40-recommendations.htm
I hate to see people slam the camera because it can't autofocus with older Nikon lenses. It is true that there is a "slight problem" with older Nikon lenses not autofocusing on the D40, but if you do not own a bag full of older lenses, it is not going to be a problem. It is barely a problem anyhow. If you check www.nikonusa.com for "AF-S" lenses, which are ALL 100% compatible with the D40, you will find 23 lenses, including 7 "VR" (vibration Reduction) lenses and one true macro lens with "VR". There are another 25-plus lenses in the current catalog that provide all functions except autofocus as well as many (possibly dozens) "out of print" lenses that will work just as well. In addition, although these lens will not autofocus, most of them will still give focus confirmation. From the D40 manual: "If the lens has a maximum aperture of f/5.6 of faster, the viewfinder focus indicator can be used to confirm whether the portion of the subject in the selected focus area is in focus. After positioning the subject in the active focus area, press the shutter release button halfway and rotate the lens focusing ring until the in-focus indicator is displayed." (See www.members.aol.com/swf08302/nikonafs.txt for a list of AF-S lenses.)
The D40 only has 3 autofocus zones arranged horizontally at the center, 9 o'clock and 3 o'clock positions. This might be considered a limitation, but realistically, most people will find this perfectly adequate, especially if you are moving up from a point and shoot with only a center zone.
There is no "Status LCD" on top of the camera, but Nikon chose to use the rear LCD for this information. This is actually a nice move, as the display is bigger than the top display and you aren't using the read LCD for anything BEFORE you take the picture anyway. This is not a step backwards and it actually makes sense to me.
You can get the D40 with the 18-55 kit lens at B&H Photo available through Yahoo! Shopping or at 1-800-622-4987) for $550 (May 2007) and this is with TWO Lexar 1 GB cards. You can get the same kit with one Lexar Platinum (higher speed) card for $560.