The main conflict of the story is Captain Ahab's hunt for the whale Moby Dick. A conflict is what causes tension in a story, and Ahab's hunt causes this tension for himself and for the rest of the crew.
The most obvious protagonist is Ishmael, since he is the narrator and the character the narrative follows; however, one could also argue that the protagonist is Ahab. Based on this interpretation of the story, the antagonist would be the whale. The whale poses the obstacle that Ahab must overcome to achieve his goal of revenge.
However, one could make the argument that Ahab is the antagonist. This is because he is willing to put other men at risk in order to take his personal revenge on the whale. In this case, Starbuck or the whale could be considered protagonists, since they are the beings who oppose Ahab.
This question is not straightforward, because Ahab is not a traditional hero. He selfishly puts the crew of his ship, including Ishmael, in danger to find revenge. This is neither noble nor self-sacrificing. Determining Moby Dick's protagonist and antagonist depends on the reader's definition of a hero.