If I were putting together the script I would try to avoid any fantastical elements beyond Thor by excluding alien invasions or cosmic powers I think.
I would incorporate morals and ethics into the reason behind needing a superpowered team. Start out the movie with SHIELD as a global peacekeeping force and intelligence organisation with UN backing and oversight. Fury heads it as public director because he has proven during his time with the marines, army, CIA and WW2 military that he is one of those officers who fights for the right reasons and isnt one of those Generals who would always have US interests at heart. Nick's a world kinda guy kinda like Cap.
SHIELD
As Rich Santoro said, SHIELD could have been monitoring various unsubstantiated events across the world, but merely tracks them at present because, well, they're unsubstantiated. A small division of SHIELD is assigned to this as a sort of futurist research group hoping to advance humanity and civilisation through using SHIELD resources and networks to study seemingly superhuman or fantastic phenomenon. Research into genetic modification of human beings is strictly prohibited in accordance with UN policies, and so the advancement of mankind is slow and semi-natural (allowing for stem cell organ replacement research and the normal drug research to extend our lives by curing or preventing disease and natural breakdown).
BAD GUYS
However, those outside of the UN (the terrorist groups, industrial billionaires with few morals in places of high influence, megalomaniac rulers etc) aren't hindered by these policies and so have managed to make some great leaps in developing their own supersoldiers, bioweapons etc. SHIELD of course strives to take out these people and prevent their creations being used. But with the sudden increase in open activity of one of the major 'evil' organisations (Hydra), SHIELD simply cannot hold back the tide any longer. Coinciding with this, a few isolated accidents are beginning to occur around the world (the US in particular) including appearances of the Hulk, the Abomination, Thor, and Iron Man - non government or UN sponsored, accidental creations of superhumans.
THE GIST
In the movie, one incident involving the Hulk up in the arctic results in the discovery of Captain America's body in the ice. This incident could either another botched capture attempt by the US army, or an attack on the meditating Bruce Banner by Hydra agents trying to grab him and manipulate the Hulk to work for them as a WMD (The abomination being incarcerated somewhere they just can't get to to use him instead). Nick Fury recognises Cap from back in WW2 when he was simply part of the Howling Commandos, and knows from what he's learned throughout his career that Cap represents THE last example of government researched and sponsored human enhancement. The timings could not be better, and Nick sees recent events as a sign to put together a team of heroes who weren't drafted, weren't experimented on and have no loyalties to any particular government (therefore making them less succeptible to being coerced by their own governments to disobey SHIELD orders etc). These heroes exist within the terms of the UN genetic modification accord, but possess the power to stop Hydra and its supersoldiers and whatnot.
During the course of the film, amongst betrayal from within (a prominent member of the UN council that oversees the team above Fury), there's the revelation that the known head of Hydra (Strucker) is actually one of a council of leaders and that their overall leader is the Red Skull who has not been seen since the War but has manipulated events around the world for decades. Further, the Skull has been putting together his own team that are powerful beyond Hydra's genetic efforts - the Masters of Evil! Members to be decided and whose purpose is to overthrow DC in decisive fashion.
HAWKEYE AND BLACK WIDOW
Brought in as specialist human SHIELD agents to reintegrate Cap to society and bring him up to speed on SHIELD protocols/procedures etc, and to train the Avengers as a team (BW could have been recruited by the end of Iron Man 2).
HANK PYM AND JANET VAN DYNE
Head of the X-filesey group, is in charge of the riviving of Cap. His fiance Jan is present and has clearance as liason for her father's company that provides funding for some of the research Hank has been carrying out in the field of inorganic shrinking for a whole host of applications. He's discovered Pym Particles (mentioned maybe by Stark when he meets Hank for the first time before Cap's revival) but so far they have proven unstable in testing. In the classic Ultimates Cap disorientation scene, the ensuing fight between Cap, Hank, and a bunch of soldiers on scene results in Hank and Jan being thrown through the cavity wall separating two of his labs and their exposure to a huge dose of Pym particles and an "as-yet-unnamed-catalyst". This stabilises the particles and causes both of them to experience size changes: Hank's initially only growth when he gets frantic when Jan appears to have disappeared in the explosion, and Jan's in shrinking in response to the explosion in an attempt to protect herself.