I am just going to pick the straight answer and say "Han." Greedo was too busy worrying about his "time" with Jabba to shoot first.

It depends which version you watch.
 
It's Greedo however.

In the original version Han shoots first, as that scene is a direct reference to the opening scene of The Good, The Bad and the Ugly. However, in later editions Lucas changed it so that Greedo shot first, to make Han seem like more of a 'hero', but it was done digitally and very poorly, in one edition Han's neck literally streches to the side, dodging the shot.

" In the original version Han shoots first, as that scene is a direct reference to the opening scene of The Good, The Bad and the Ugly. However, in later editions Lucas changed it so that Greedo shot first, to make Han seem like more of a 'hero', but it was done digitally and very poorly, in one edition Han's neck literally streches to the side, dodging the shot. "

Han shot first.

Lucas shot first, and second, and third...

I'd say Greedo was intended to shoot first even if that wasn't the case in the original cut.

"In the original version Han shoots first, as that scene is a direct reference to the opening scene of The Good, The Bad and the Ugly. However, in later editions Lucas changed it so that Greedo shot first, to make Han seem like more of a 'hero', but it was done digitally and very poorly, in one edition Han's neck literally streches to the side, dodging the shot. "