Strictly speaking, S.H.I.E.L.D. didn't exist during the time of Captain America: The First Avenger, but its precursor certainly did. The Strategic Scientific Reserve -- known as the S.S.R. -- was directly responsible for the creation of Captain America. Unfortunately, they are also somewhat responsible for the death of Dr. Abraham Erskine, who designed Project Rebirth. Despite all of their safety protocols, a Hydra agent managed to slip through their safeguards, and he assassinated Erskine just as Steve Rodgers emerged from the chamber. With Erskine dead and the formula lost, the S.S.R. was unable to recreate the process that resulted in Captain America, failing its goal to create an army of super soldiers.
At first, the S.S.R. wouldn't even send Captain America into battle, using him only as a figurehead to sell bonds to support the troops. However, when several S.S.R. agents -- including Steve's best friend Bucky Barnes -- were captured by Hydra, Cap took matters into his own hands. With some help from Agent Peggy Carter and scientist Howard Stark, Rodgers then launched a military campaign that brought Hydra to its knees. He also singlehandedly wrested the Tesseract (which we now know as the Space Stone) from Red Skull and put an end to the Hydra leader's reign of terror.
Later on, the S.S.R. became S.H.I.E.L.D., the organization that found Rogers trapped in the ice and resuscitated him. Per S.H.I.E.L.D.'s usual M.O., the organization wasn't exactly straight with him when he woke up, opting to pretend as though he was still in the 40s in order to make the transition easier for him. The deception didn't last long and caused quite the scene when he burst out of his confinement, running right into a crowded Times Square. At that, Fury was forced to reveal the truth.
S.H.I.E.L.D. had perhaps its biggest role in The Avengers, which brought the protagonists of Iron Man, Captain America, Thor and The Incredible Hulk together on screen for the first time. When Loki stole the Tesseract from S.H.I.E.L.D., Fury realized what kind of threat that power posed and kickstarted the Avengers Initiative. However, he was less than forthcoming about S.H.I.E.L.D.'s Tesseract research; he told them they were looking for an unlimited power source, while they were actually building weapons for the eventuality that Earth would be invaded by an alien force. As Tony Stark put it, S.H.I.E.L.D. immediately pursued a "nuclear deterrent" -- and then lied to them about it.
Fortunately for Fury, the proto-Avengers didn't have very long to argue the point, because Loki launched an attack on the helicarrier shortly thereafter. Taken unawares, S.H.I.E.L.D. didn't exactly provide much of a countermeasure. While Fury and his right-hand woman Maria Hill were able to hold their own, the end result left the Avengers scattered -- and Phil Coulson dead. In a rather manipulative move, Fury then used Coulson's death as a tool to unite the team, even going so far as to invent a circumstance to make it seem even more tragic.
For what it's worth, Fury did rail against the World Council when they attempted to nuke New York City. In fact, he did everything in his power to prevent it from happening, even though he couldn't do much with the helicarrier -- and subsequently S.H.I.E.L.D. -- hobbled by Loki's earlier attack.