Difference between 关 and 管 in the context of caring?
While all three can be translated as "I don't care" from time to time, there are still subtle differences. 

"我不关心" typically means "I am less/no interested in something". For example, "我不关心政治" ("I don't care about politics").

"我不管" is typically used in the case where you insist on something despite the constraints and/or the potential consequence. You acknowledge the presence of the constraints/consequences, but you care less about them. For example, "我不管老闆怎麼想，我下週就是要請假" ("I don't care how my boss thinks. I will take vacation next week.").

You can also reguard “我不管” as “No matter what happens, I must do something……”. And this is usually used in some senarios that is bad or not very good for you, and you have something mood of 破罐子破摔 (Bad as it is). So this is something in the extreme situation.

Please see this dialog：

妈妈：天冷了，别出去了。

我：我想看外婆，外婆病着呢。

妈妈：那你打一个电话问候一声吧。再说你这几天自己也不舒服呢。

我：不管了，外婆病重得很，我一定要去看！

For the 2nd meaning：我不管Something（This means I'm not reponsible for something, it has nothing to do with me.）

"我不在乎" is similar to "我不管", but in the cases you care far less about the constraints and/or the potential consequence. For example, "我不在乎花多少錢, 請醫好我兒子" ("I don't care how much it costs. Please save my son."). This is much less in mood, compared with 我不管, a neutral word used to mean that you are trying your best to do something with any possible solution.

For the 2nd meaning：我不在乎 sometimes also means “我不介意，不关心某事” (This usually means something happening doesn't effect on me seriously or in faulty).

You can understand this in the senario：

爸爸：你为何政治考试不及格？

学生：反正政治是副科，无所谓的，我不在乎。(I don't care about whether I could pass the exam of politics or not because the lesson isn't a major one for me, and on the other hand, it cannot make me repeat my course again).