I want to see some gameplay at E3 and take a look at the mechanics but hopefully we catch a glimpse of the story at E3

I don't know if they'll release any parts of the story, but I feel like they may go the route of introducing one of the side missions available early on and do a couple parts of that mission.

Judging by Bethesda's Fallout 3 and Skyrim presentations, I don't think we'll learn about the specifics of the story at E3.

I honestly think that at e3 they're just gonna show some clips of the game and some key features, but they might spoil it and make us wait until the game comes out to fully experience

Are you talking about Fallout 4? If so they'll show gameplay. If they show any part of the story people will be upset that it's a spoiler gameplay footage.

They'll more than likely demo a part of a quest and probably explore the wasteland a bit as a chance to demonstrate revamped and/or new mechanics.

I doubt we'll see much of the main story. We may get like, a blurb about it though. Or hints towards it. Like with Skyrim being told you're the Dragonborn and you're trying to stop an ancient evil or something to that effect. Or New Vegas like "You're trying to find the guy that shot you". Something small and not very descriptive like that.