I would say the East Side of Providence. Wayland Square area has a decent amount of families and is walkable to a few parks. Also walkable to grocery stores, restaurants, parks, etc. There are also part-time preschools that you can walk to as well. Depending on traffic, 10-15 minutes to the hospital.

Source: we moved there with a newborn and had our second child on the East Side. We lived off Butler Ave. We moved to the suburbs for better schools but really miss the East Side.