I know plenty of people who have kids but no car. My best friend and his 2 toddler sons lived with me for 2 years and we had no cars, and no problems. We did shopping and Dr. appointments on the bus, or walked to the store and cabbed home. The older boy went to pre'school on the bus, just like the neighbor kids. One of our favorite activities was strolling along the river bank to a playground about 2 miles away. Matt-matt rode on his little two-wheeler with training wheels and the younger Dusty got pushed in the stroller.
This was several years ago, before I rode a bike, so I'm afraid I don't have any tips for integrating bicycles into family life. But carfree with kids is certainly possible if you live in a good area.
One person wrote that big cities are impractical for carfree living. I think most here would agree with my experience that dense cities (regardless of size) are actually among the best places to live carfree.