In "Hamlet" when Hamlet finally takes action, who gets killed? Why?

The short answer:  Everyone.  Granted, before he took "active" action, Polonius and Ophelia died; they died as a result of his rash decisions, either to put on an "antic disposition," or to act quickly in his mother's quarters, without even thinking about who might be behind the curtains.

It is after Hamlet is sent away, to war, and sees the bloody scene before him on the battlefield that realizes that he's been a coward, "unpacking [his] heart with words," and needs to actually act upon his resolve.  And, intercepting the letter given to his buds Rosencrantz and Guildenstern certainly helps; he has pretty definite proof of Claudius's murderous intentions.  So, he takes action.  Rosencrantz and Guildenstern, the unfortunate bumbling friends, are killed.  He gets home, and by that time Claudius has sneakily convinced Laertes to challenge Hamlet to a duel.  In the course of the duel, the king, queen, Laertes, and Hamlet himself are killed.  At the end of the play, it is only the faithful Horatio that remains alive, to tell the tale of the unfortunate Hamlet.

The real question is though--does Hamlet actually ever take real action?  I would say that he did in "dispatching" Rosencrantz and Guildenstern, but, the death of everyone was not a direct result of his action, but rather his inaction, his antics as he hummed and hawed over whether or not to kill Claudius.  His inaction and bizarre behavior in the meantime throw everyone off, and unleash a chain reaction that leads to the duel.  In accepting the challenge to the duel, Hamlet is merely going along with the plans of others, as usual.  Beforehand, fortunately, he had finally come to a place where he could say, "the readiness is all," and he used that readiness to kill his uncle at the duel.  His readiness helped aid in that, for sure.  I hope that those thoughts helped ;good luck!