There are clearly a number of events of great importance that occur in this novel, however, if forced to select five, I would pick the following:
1) The event whose importance echoes throught the remainder of the novel is when Gene jiggles the tree so that Phineas falls off and shatters his leg.
2) The way in which following this episode Gene almost "becomes" Finny through beginning to excel at athletics is worthy of attention.
3) The winter carnival that Finny organises and holds is very important in terms of relating Finny's fantasy world to the real events of war that are going on.
4) The mock inquiry that Brinker holds where Gene is accused of deliberately jiggling the tree is clearly important, combined with Leper's experience with the army.
5) Finally, Finny's death and the rich symbolism that this contains is clearly an incredibly significant section of the novel.
I would like to take a little bit of a different approach to your question. I find it interesting that you specifically mention "five" events because that number corresponds exactly with the five main elements of plot. As a result, I would like to approach your question by accessing the elements of plot in this novel: exposition, inciting incident (conflict), rising action, climax, falling action/resolution.
Although I wouldn't say any one event is "important" in the exposition, it is significant to learn about the friendship between Gene and Phineas as well as the school they attend. This is all part of the exposition. Secondly, the inciting incident (and indeed the conflict) in the story is when Gene shakes (doesn't shake?) the tree in order for Finny to fall off and hurt his leg. This truly begins the rising action of the novel which involves Gene's sudden interest in athletics, the winter carnival, and even the mock trial that accuses Gene. The climax, or the height of the tension, is definitely Finny's death. Finally, I would say that there is a falling action, but not necessarily a resolution to this plot. There is no way to "resolve" what happens to Finny. But Gene does look back upon the events and loss of innocence as part of the falling action.
So the more things remained the same, the more they changed after all. Nothing endures. Not love, not a tree, not even a death by violence.
In this way, exactly "five events" constitute the plot structure of the story. As a result, it is a worthwhile novel to use for a teacher to instruct students on the elements of plot.