Even the 18-55 is a great lens for baby pix, and the camera will be a big step past a P&S for quick reaction. It will be a great learning lens and has an excellent zoom range for basic snapshooting.
The mid-range zoom is much more flexible than the 35mm, which though an excellent lens IMO has a relatively narrow field of view, very restrictive particularly for shooting inside (though the additional light-gathering power is quite useful inside in available light). My "normal" shooting is typically at 16-18mm or so or 50-75mm, rarely use the mid-range of my mid-range zooms, which is where the 35mm puts you. Shooting kids is more than shooting portraits - it's their activities, their environment, their friends and family.
My suggestion for the same amount of money as a step-up lens is the SB400 flash, $120 or so, which will be a big improvement over the built-in flash, allowing you to use indirect flash for much more natural flash lighting. If your budget permits, the SB600 or SB900 flashes plus off-camera bracket would provide even more lighting flexibility, at a cost in weight and bulk.
Enjoy - this camera should substantially reduce your frustration with shooting delay.
However, you should be aware it will bring a whole new range of frustrations you will become aware of, including lens speed, AF speed and tracking, lens focal length range, equipment weight, storage, accessory carrying, lighting, etc. (I think everyone here will agree that to be always ready for baby bix, you really need to carry both a D3s and D3x, with 14-24, 24-70 and 70-200 lenses, 85mm f/1.4, 35mm f/2, 100mm f/2.8 AF-S micro at a minimum. Each room of your home should of course be set up as a studio, with multiple flashes, flash controllers for each camera, a beauty dish, two or three softboxes, C-stands, reflectors, etc, to be always ready to shoot the best baby pix in every situation.)
You'll also want to have a good P&S for those times when you don't have this bulky stuff with you.