First things first. The movie employs two levels of narrative trickery, which can make things very disorienting at times. They are:
So anyway. Forrest Gump is a mentally handicapped boy who lives in a decaying Southern mansion with his mother, who seems to be a few eggs short of a dozen. She is destitute, and is forced to supplement her meager income with prostitution. Basically it's Part One of The Sound and The Fury. On some level, Forrest is aware of what his mother is up to, and this has warped him. Consider the scene where she has sex with the principal and Forrest imitates their noises as he leaves - in the movie as presented, this is jarring and out-of-place, but it fits perfectly with this interpretation.
Forrest's mother, in a state of denial about her son's condition, makes the disastrous decision to send him to a regular school. Keep in mind, Forrest isn't just "a little slow", he has some real psychological problems. There, he is picked on regularly, and never gets the special help that he needs. He's unable to compete in his classes or relate to his peers, so he retreats into fantasy. (He imagines that he met Elvis, for example.) He does manage to make one friend, Jenny, because her background is even more screwed up than his.
As they get into the high school and college years, Jenny does something that horrifies Forrest - she becomes a woman. Forrest's madonna/whore complex causes him to become obsessed with her "purity" to the point where he attacks any boy who comes near her. Jenny had tolerated Forrest up to this point, but the scene at the women's college is her breaking point. (He imagines the next scene where he feels her up. Remember: unreliable narrator.) Jenny transfers to another college in a different state and leaves no forwarding address.
After high school, Forrest works a series of menial jobs, such as landscaping. His fantasy life becomes more involved - he sees a Crimson Tide game on TV, and imagines that the sign says "GO FORREST". He watches the news a lot, and the rising tide of the sexual revolution upsets Forrest to the core. He projects all of the excesses of the era onto his memories of Jenny. (He doesn't actually see her during most of these scenes.) Occasionally you see reality creep in - the stack of letters marked "return to sender" for example.
The news gives him two other things to fixate on, and they're both pretty disturbing: Assassinations, and the Vietnam War. He even goes so far as to construct a Vietnam scenario for himself, with scenes cribbed from various war movies - Notice he was supposedly wounded, but suffers no lasting effects, PTSD, anything like that? After his mother dies, he is homeless for a while - his cognitive dissonance forces him to justify it with a story about how he "went running", but it's a thin story and he can't really explain it.
All of this sets the stage for the chilling final act. After sitting on a park bench for hours telling strangers his rambling, incoherent "life story" (in which he shows people a copy of Forbes with Lee Iacocca on the cover, and tells people it's him) he goes on a psychotic break. He has managed to track down Jenny, and he kills her and abducts her son. Of course, in his mind, he has whitewashed this as her dying of that mysterious virus he saw on the news, but she certainly didn't look sick...