My kids' first pet was a cat that my husband and I adopted before we had children. The cat was great with the children when they came along - he had a personality more like a dog, and would follow them around and was very loyal. Never scratched or bit, even when they were a little too rough with him. He lived till about 16 or so - and the kids were just heartbroken to lose him. Since then, we've had a couple more cats (one died very young) and eventually got a puppy for them (actually it was more for me), but the funny thing about the dog is that while she likes me and the kids, she is much more bonded to my husband, who was never a "dog person." I think it's great for children to have pets, but they are a big commitment!
P.S. The only pets I had as a child were a pair of gerbils - we thought we got 2 boys, until one of them had a litter and proceeded to eat them all! Really gross. At least cats and dogs don't do that!