In the broader scope of the Marvel Cinematic Universe (we're currently twelve movies down, at least nine more to go), the most important influence Captain America: The First Avenger has on the plot is bringing the beacon of righteousness from his 1940s origin to the modern day ahead of The Avengers.
With that in mind, the film pretty much gives you everything you need to know ahead of the big team up in the first five minutes. Seriously - after the Paramount logo (hard to believe now the franchise was once not a part of Disney), the film has a brief, four minute sequence showing a bunch of S.H.I.E.L.D. agents discovering the frozen body of Cap. Roll credits.
Obviously it doesn't give you any of the cool period stuff, nor the background to Hydra (although as they fill generic evil group status there's not all that much you need to know), but if we're taking the MCU as its own super story (which it pretty much is at this point), The First Avenger is one of the easier to skip in a mega-franchise rewatch. Although don't do that - it's the best of the pre-Avengers films.
What other films give the entire plot away at the beginning? Share any examples we missed down in the comments.