I'm pretty sure that if a creature is enchanted with some other aura, say Rancor, and then you put Gift of Immortality on it and the creature dies, Rancor falls off when it comes back to the battlefield. Is this true? My instinct is yes because the creature goes to the graveyard and becomes a new object and all that but I'd like to be sure. As an added question would Rancor's and Angelic Destiny abilities work off a creature that dies while enchanted by Gift of Immortality?
After a creature dies, but before triggered abilities are placed on the stack, State Based Actions are performed. The SBAs will cause both Rancor and Gift of Immortality to go the graveyard because they are not attached to anything [CR 704.5n].
Then, it's time to place on the stack the abilities that have triggered. Gift of Immortality's ability and Rancor's ability are placed on the stack in whatever order you choose.
When Rancor's ability resolves, it returns to your hand. When Gift of Immortality's ability resolves, it returns the creature to the battlefield and creates a delayed triggered ability, which puts Gift of Immortality back on the battlefield, attached to the creature at the beginning of the next end step.
Angelic Destiny's ability works the same way Rancor's does, so it would also return to your hand.
When a creature dies, all Auras enchanting that creature are moved to the graveyard. That includes Rancor, Gift of Immortality, and Angelic Destiny. Then, triggers go on the stack. In more detail:
A creature enchanted by Rancor dies. Then, Rancor goes to the graveyard. Then, Rancor's triggered ability returns Rancor to your hand.
A creature enchanted by Angelic Destiny dies. Then, Angelic Destiny goes to the graveyard. Then, Angelic Destiny's triggered ability returns Angelic Destiny to your hand.
A creature enchanted by Gift of Immortality dies. Then, Gift of Immortality goes to the graveyard. Then, Gift of Immortality's triggered ability returns the creature it was enchanting to the battlefield. Then, at the beginning of the next end step, Gift of Immortality is reattached to that creature.
If all three are enchanting the same creature, and that creature dies, then you get to decide the order in which to stack these triggers. However, the effect will usually be the same regardless of the order in which they resolve.