(Here are some posts on my own blog that discuss openings.) To keep this relatively short, I want to focus on the all-important last paragraph.
The beginning of a story has a lot to do, and it might be most helpful to write your opening, write the rest of your story, then come back and revise the opening so it is more effective in setting up the plot questions and themes. I was helping a friend with a story just today, and we discussed the "end of the beginning." This book is about a girl raised in Europe who was forced by her parents to study piano for years. She is disillusioned by music and eager to get far away from her parents, so she chooses a college in the US that has lost its music program. That's the opening, setting up her college story.
I suggested that the author think about what is going to happen later in the book. The college is going to resuscitate the music program and recruit the protagonist to be the first major, and in the end of the book she's going to found her own punk band, showing that she has chosen her own way (not the parents or school). Boy! This is good, because it forces her to change, to learn to value her own talent, to choose rather than just react.
The end of the opening, however, could set up the "praxis" of her journey, by posing a bit of a conflict or question. In a way, the last paragraph in the opening could serve as a "hinge" to the rest of the story, actually helping to open up to the rising conflict and rising action of the middle, and hinting at the theme that will be resolved in the ending.
His first chapter has her choosing a college, deliberately selecting the one that has lost its music program. I suggested a final paragraph that would emphasize what he wants the reader to think about. But to achieve that, he must identify what that is! Does he want the reader to think about her disorientation at being in the US after Europe, a fish out of water? Or her sense of her musical talent being trapped by the expectations of her parents even as she arrives in this new place?
He agreed with the latter, that her journey should start with her resistance to those expectations, and so he wanted to draw the reader's attention to this. So he ended the first chapter this way, "My first class was History of Culture, in the Humanities Quad. Shoved into a corner of the lecture hall was a grand piano, swaddled in a gray quilted cover. I hurried past and took a seat in the center, directly in front of the professor."
This sets up the conflict between her desire to be "merely a student" and her musical talent and provides a concrete action (hurrying past the piano) to symbolize the beginning of her journey from resistance to self-acceptance. If the author wanted to emphasize her "fish out of water" aspect, how could that be achieved with the same situation (entering her first class lecture hall)?
Another way to use that final paragraph in the first chapter is to set up a motif (a recurring thematic image or concept) which the rest of the story will develop. For example, in my Regency novel Poetic Justice, the first chapter pits the hero John against an enemy, who tries to trick him by offering an alliance and then trying to kill him. I was worried that the adventure of this opening would conflict with the quieter aspects of the rest of the story. But when I realized that no matter what the situation, John was always being "tested," especially by the class system that scorns him as a tradesman.
By the time his shipmates arrived panting, daggers drawn, the light was gone entirely and the dock was slippery with blood. Two of the bandits had fled, and the third lay unconscious on the dock. John loosed his death grip on the saddlebag, let his first mate take it, let his steward peel his fingers from around the knife and put it away. He nudged the bandit with his foot. "Tell your employer," he said, then paused to drag in a breath, "that I passed that test too."
Thus, in the final paragraph of the first chapter, I emphasized this motif to connect this scene with the rest of the story, which develops and finally resolves the recurrent pattern.
Look at your own first chapter and think of how you might use that last paragraph to wet up the rest of the book, by establishing the context or conflict, by posing a question the rest of the story will answer, or by connecting the first scene with the rest of the story using a theme or motif. Any examples from your work?