I'm a little unfamiliar with how things go on the stack, I haven't played competitive magic since 1997, so this stack business is new to me. Having said that: can I cast windfall, let it resolve and then in response draw a bunch of card's via yawg barg in order to make my opponent draw enough cards to kill him with underworld dreams on the table? My concern is that I don't want to cripple myself only to have the spell countered. What is the timing here? Do I have to respond with Yawg Barg before my opponent has a chance to respond or can I wait until they let the spell resolve before I respond?
In addition to what Puck said, once a spell resolves (in this case Windfall), during the resolution, you can't play activated abilities (such as Bargain's ability). If there are any triggered abilities, they would go on the stack AFTER the complete resolution of the spell.
So if you cast Windfall, then pass priority, and your opponent passes priority back to you, Windfall would start resolving (no Bargain).
The only way to be sure if they aren't going to counter the Windfall is if they pass priority back to you and Windfall starts resolving - once you start activating Bargain, the opponent will have priority between the resolution of the Bargain activations and before the Windfall starts resolving, so they would be able to counter it.