For more extensive recommendations, you might want to purchase a copy of Frommer's New York City with Kids, an entire guidebook dedicated to family visits to the Big Apple.
Good bets for the most timely information include the "Weekend" section of Friday's New York Times, which has a section dedicated to the week's best kid-friendly activities; the weekly New York magazine, which has a full calendar of children's events in its listings section; and Time Out New York, which also has a weekly kids section with a bit of an alternative bent. The Big Apple Parents' Paper is usually available, for free, at children's stores and other locations in Manhattan; you can also find good information from the folks behind the paper at www.parentsknow.com.
The first place to look for babysitting is in your hotel (better yet, ask about babysitting when you reserve). Many hotels have babysitting services or will provide you with lists of reliable sitters. If this doesn't pan out, call the Baby Sitters' Guild (tel. 212/682-0227; www.babysittersguild.com). The sitters are licensed, insured, and bonded, and can even take your child on outings.
Recommended family travel websites include Family Travel Forum (www.familytravelforum.com), a comprehensive site that offers customized trip planning; Family Travel Network (www.familytravelnetwork.com), an online magazine providing travel tips; TravelWithYourKids.com (www.travelwithyourkids.com), a comprehensive site, written by parents for parents, that offers advice for long-distance and international travel with children.