The 602Z is indeed a very good camera. looking at your needs (taking children photo especially) you will need a fast camera, with fast processing, and very low shutter release lag. In this regard, the 602Z is the current leader...
However, think about getting the following accessories, in the short to long run:
SHORT TERM
Immediately get extra sets of batteries and a good charger ($100 should get you 3 sets of AAs and a decent charger)
Extra memory cards (at least 1 piece of 128MB)
Suitable camera bag (consider space for extra batteries)
LONG TERM
Tripod
External flash
At a budget of 1500, you have many choices. Perhaps a minimum of 3 megapixel (more than enuff for 4R prints as imaginary_number stated), at least 3X optical zoom, fast processing, and a hotshoe.
5 MP Contenders:
Sony DSC 717 (big, need basic photography skills to maximise potential, proprietary batteries, proprietary memory cards) ard. $1600
Nikon 5700 (currently hot-seller, may be tough to get at under 1500, propreitary batteries) ard. $1600
Oly 5050Z (currently cheapest among the other 2, use 4 AAs) ard. $1200
4 MP Contenders:
Canon G2/ G3 (similar to S45, except for swivel lens and hotshoe, however, focusing may not be as impressive as the Fuji 602Z, proprietary batteries) ard $900 (G2)-$1200(G3)
Canon S45 (very compact, processing quite impressive, good colours, bring everywhere, proprietary batteries, not hotshoe) ard $900
3 MP Contenders:
Finepix S602Z (fast, sharp, use 4 AAs) ard $1000
Canon S30 (slightly outdated, proprietary battery) ard &700
Nikon 885 (compact, sharp images, proprietary battery) ard $700
Personally, I will get a $1000 camera, and use the remaining money to buy accessories, which can be quite expensive in the long run.