Directly related to "care":
* Carefree: Without worries or responsibilities
* Careless: Showing a lack of attention or concern
* Caregiver: A person who provides care for another
* Caring: Feeling or showing concern for others
* Carefully: With great attention to detail or safety
* Caretaker: A person who looks after something or someone
* Careworn: Showing signs of worry or fatigue
Derived from "care" but with different meanings:
* Uncaring: Lacking empathy or concern
* Discare: (Archaic) To neglect or abandon
* Miscare: To take care of something improperly
Other related words:
* Concern: A feeling of worry or interest
* Attention: Focus or consideration
* Concernment: (Archaic) A feeling of interest or worry
* Tenderness: Kindness, gentleness, or love
This is not an exhaustive list, but it provides a good starting point for understanding how the root word "care" can be used to create different meanings in different contexts.