Nurse, nurture, foster, cherish, cultivate are comparable especially when they mean to give the care necessary to the growth, development, or continued welfare or existence of someone or something.
Nurse basically implies close care of and attention to someone (as an infant or a sick person) unable to care for himself with the idea of helping that one to grow strong and self-sufficient. In extended use the term implies similar sedulous attentions that feed or nourish and thereby strengthen what was at first weak, indefinite, or tentative.
Nurture stresses the rearing and training, and so the determination of the course the person, or by extension the thing, will follow.
Foster implies encouragement or promotion of the growth or increase of something.
Cherish stresses loving, protective care (as of a nurse or a parent for a child, or of a husband for a wife). In its extended use it is not always distinguishable from nurse, but it may retain its implications of holding dear or as a thing of value, and stress prizing and preserving rather than brooding over or causing to increase in strength.
Cultivate basically implies the care and attention given to land in order to increase its fertility or to plants in order to improve their condition. In its extended use it implies comparable and equally sedulous attentions to the improvement or growth of some usually desirable thing.