To know the score means to know the facts about life and its difficulties, but in this context the speaker is saying she never got involved with people who were naive to know about pleasures of life and its beautiful moments but not life and its difficulties
Consider the staetement "I never got involved with anyone who was married or too young or too naive to know about the pleasures of life and its beautiful moments.". Based on this statement, is "I never got involved with anyone who was married or too young or too naive to know the score." a true statement? Yes/no/unknown