Develop a sensitivity to the 'voice' telling the story. From whose perspective is the story being told? Is there a first person narrator? How does the way in which he or she describes events reflect his or her perspective and attitudes, and is his or her narration 'reliable'? If the story is told in the third person, does it focus on describing the thoughts and feelings of one character, does it only describe external actions, or does the author allow us to 'see inside' the thoughts of many different characters?
You must be able to visualize the 'flow' of a story, understanding where the character(s) go, where events happen, how the characters and the plot develop, and the themes within the story.
It is helpful to construct a flow chart after reading showing prominent characters, where they go, what happens to them, how they change (for this you must look very carefully at their actions and dialogue and ask yourself 'Is that what they would have done/said at a previous point in the story?), and what events caused that change.
The most important thing to read for is character development, where you analyze characters to see how they have changed over the course of the story.