에게 is used in writing.
친구에게 전화했다. I called my friend.
동생에게 선물을 줬다. I gave my younger brother/sister a gift.
너에게 실망했다. I was disappointed by you.
너에게서 희망을 보았다. I saw hope in you.
한테 is used in speech.
친구한테 갔었어. I've came to my friend.
동생한테 전화왔었어. I got a call from my younger brother/sister.
(한테서 is the original form, 서 is left off.)
며칠 전부터 잠을 제대로 못 잤어. I couldn't sleep well since a few days ago.
여기부터 시작하자. Let's start from here.
너부터 해. You do it first.