With how they've set it up, with Fury being what unites them, and probably with S.H.I.E.L.D. backing them, I'd say the easiest and most likely story (or at least what I'd do) is some sort of Ultimates based story. An impending alien invasion the team has to fight off. a simple enough but threatening obstacle tha could let all the avengers shine in battle. Make Baron Zemo take the place of Kleiser (either make him an alien, or have the aliens bring him back to life to lead them once more). Basically Ultimates 1 vol 2, skipping the first story with Hulk as the bad guy. Not sure about part 2, but for the 3rd one, have Loki from Thor's solo movie show up and unite all the seperate bad guys to attack NY at once, essentially doing Ultimates 2. He is joined by Abomb, Whip/dynamo from IM2 (only a much bigger threat, like Dynamo from Ultimates 2), Sterns from TIH (now mutated into leader) to mind control the Hulk into attacking his teamates (using Avengers vs Hulk from Ultimates 1 without making hulk the bad guy), etc. We get a HUGE epic fight for the 3rd film, including the asgardian trolls and vikings, which would help even more to tie in our's and Thor's world. Biggest Superhero fight EVER on screen. (disregard the black widow being a bad guy and hawkeye killing her aspect. It was awesome in the books, but I don't think it's neccessary to have that aspect in the movie, nor do we really need hawkeye's family).
At the end, we see the Avengers decide to dissconnect from S.H.I.E.L.D. in order to be able to help the world without it being seen as US government action.
Now that the team would simply be a public gathering of heroes, start to gear more toward 616 avengers, bringing in more characters from the classic avengers tales, both heroes and enemies. Falcon, She-hulk, whoever. It frees it up to go back to classic stuff because you have the core team established, and they are no longer owned and operated by Fury, though it'd be nice to see him pop in from time to time to ask for their help or whatever.
I know some people don't like the ultimates because they find the characters to be "jerks", but the easy answer to that is to just not write them as jerks. I've never read any 616 Avengers stories, but if that's the favored characterization, personality-wise, use that. The story of the Ultimates can still be used in the broad sense of what the plot and villains are.