Depending on the size of your city, this may not be possible. I don't live in a hugely populated area. We have a few pediatric practices, and I know more medical professionals now than I did when we lived in Dallas-Fort Worth.
When your child is sick, I mean really sick, is it beneficial to know your child's doctor socially? I do. But I worry about taking advantage of that relationship. When is it OK to call his or her home number? How sick does your child need to be?
This etiquette was not taught. I don't have a good answer. But if my child doesn't improve markedly soon, I may just see how friendly the doc is when he gets an early morning phone call.