The FW zoo is very nice! The Log Cabin Village is close by and cute. You will most likely love making the hour drive out to Glen Rose for Fossil Rim http://www.fossilrim.org/drive.php
and they have the Dinosaur Valley out there too. Like someone said, FW Child is a free publication at many stores and there is another one called "Suburban Parent" that lists calendar of events for families. They do the cattle drive daily through the Stock Yards. The longhorns are pretty impressive to see.
FW also has many story times at local libraries and book stores. Trinity Park is pretty nice and has a big duck pond. The Botanical Gardens (free) and Japanese gardens are both nice. FW is home of world class art museums, most free, and my kids like to go see the art from time to time.
FW also has events/festivals almost on a monthly basis around town. The StarTelegram (paper) puts out the entertainment guide each week (I think it's in with the Friday paper). Check out Burger's Lake for summer swimming. http://www.burgerslake.com/
. Sundance Square is very safe and clean. Fort Worth Nature Refuge is a short trip west of town http://www.fwnaturecenter.org/
. The Gaylord Texan (in Grapevine) often has some neat things for the kids to see and do, especially around the holidays. Sandy Lake is near Farmer's Branch/North Dallas but it's a cute, smaller amusement park that has more rides for smaller children than 6 Flags http://www.sandylake.com/
. That's all I can think of at the moment. Hope that helps...