Pedaltexas.com was a great resource but it's not around anymore.
This site lists a lot of trails and gives some info on their difficulty.
http://www.texasoutside.com/bikeTrail.html
The Dallas Off Road Biking Association maintains most of the mountain bike trails in North Texas. Some are tougher than others, but most have sections that are ok for kids. Loops 1 and 2 of Rowlett Creek Preserve are a good place to start.
http://www.dorba.org/
River Legacy Park in Arlington has some nice paved trails and a new mountain bike section.
The greenbelt from Hwy 380 east of Denton up to Lake Ray Roberts is nice and flat.
Duck Creek in Garland is ok but probably not worth the drive for you - you're better off with River Legacy.