In an experiment to see if having a nurse come three times to visit the homes of newborn babies reduces doctor visits, what is the independent variable

An independent variable is a variable that can be changed or controlled. A dependent variable on the other hand is a variable that you cannot control. If you look at your problem, you cannot control doctor visits. In this case, it can be categorized as an effect of the experiment, which is a characteristic of a dependent variable. 

The number of nurse visits however, is the controllable variable. It is the cause of the change in the dependent variable. So your independent variable is the nurse visits.