1. Opponent plays Garruk
2. Opponent activates Garruk
3. You lighting bolt your opponent and redirect it to Garruk.

As long as your opponent didn't do anything else between playing Garruk and activating him then he retains priority and the ability will go on the stack.
If your opponent goes to play something else before activating Garruk's ability; another spell or ability, then it's generally understood that they have passed priority (I'd make sure everyone understands this in your group before making an issue about it).
Before that ability can resolve he would need to pass priority to his opponents and they can then respond to the ability, like with your Lightning Bolt.
The Lightning Bolt then goes on the stack.
If there are no other responses and everyone passes priority the stack resolves.

Your Lightning Bolt would deal 3 damage to Garruk and he would lose 3 loyalty counters.
Whether Garruk would die is dependent on whether your opponent used a + or - ability. If your opponent added loyalty then Garruk would survive, because activating a planeswalker's loyalty is a cost and players are unable to respond to those (just as shwin mentioned).
If he took away loyalty then Garruk would die and be put into your opponent's graveyard due to 0 or less loyalty (state-based effect is the general term from what I recall).
Then Garruk's ability would resolve and do whatever it is.

Another good example of this is with Sarkhan, Dragonspeaker:

Player A: Plays Sarkhan. No responses from either player, so Sarkhan enters the battlefield.

Player A: retains priority and activates Sarkhan's +1 ability to turn him into a 4/4 indestructible hasty dragon. Then passes priority.

Player B: Plays Hero's Downfall in response to Sarkhan's ability. No other responses.

Sarkhan will die before he becomes indestructible since his ability doesn't resolve before the Hero's Downfall to make him indestructible.