The ability checks the targeted creature's power twice: when the creature becomes the target, and when the ability resolves. Once the ability resolves, it will continue to apply to the affected creature no matter what its power may become later in the turn.
Lethal damage and effects that say "destroy" won't cause a creature with indestructible to be put into the graveyard. However, a creature with indestructible can be put into the graveyard for a number of reasons. The most likely reasons are if it's sacrificed, if it's legendary and another legendary creature with the same name is controlled by the same player, or if its toughness is 0 or less.