These three verbs are easy to mix up, but they aren't too difficult if you spend some time learning each word's function. It is of course more complex than this page allows, but this should give you a fairly good understanding.
The Word: あげる
When you, the speaker, give something to someone, use あげる
kore o anata ni agemasu.
I will give you this.
NOTE: The receiver is the one with the に particle (あなたに to you).
The Word: くれる
This is also usually translated as 'give' but it is from the receiver's point of view.
tanaka san ga kore o watashi ni kuremashita.
Tanaka gave this to me.
NOTE: Again, the receiver is the one with the に particle. (私に to me)
The Word: もらう
もらう is used from the perspective of the receiver.
watashi ga tomodachi kara ke-ki o moraimashita.
I received a cake from a friend. (My, the receiver, point of view)