There are two common ways to run a game based on the world of comic books and four-color action: the episodic story and the ongoing story.
Episodic stories are generally open and shut. The heroes are going about their business when a villain appears and causes trouble. There's some difficulty in overcoming the villain at first, but the heroes eventually triumph! The villain usually ends the story imprisoned or otherwise defeated, but can return for future issues. There can be recurring story elements and supporting characters, but they are usually not the focus.
Ongoing stories are characterized by their story arcs. These stories can braid together and interact - for example, the story of two characters' attempts to reconcile their relationship can intersect with a villain attack on a major city.
There's no "right" or "wrong" way to run, and your group's style might not match either of these two approaches.
The GM can create genre stunts (see "New or Optional Rules") or introduce other rules that enforce the flavor they are interested in promoting.