From the beginning of the text to the discovery that the Time Machine has been stolen (the "Dude, Where's My Time Machine?" moment).
In Act I of The Time Machine, we're introduced to the Time Traveller and his doubting contemporaries (and the narrator, who is less doubtful). The Time Traveller makes a case for time travel, then just a week later he claims to have done it. He starts telling the story about how he went to the future, saw the landscape change along the way, and stopped in a Utopian-seeming age. The future people, the Eloi, are beautiful, but stupid and lazy; they don't do any work and just play all day. (To be fair, that's how we like spending some days too.) The Time Traveller theorizes that the people have become weak because they have machines to do their work for them. And then his Time Machine gets stolen.
From the loss of the Time Machine to the trip to the Palace of Green Porcelain.
When the Time Machine is lost, the Time Traveller is truly screwed. He has no choice but to stay and explore the future. The Time Traveller discovers that the Eloi are haunted by the people who live under the ground, the Morlocks. He follows them down to their caverns and discovers that they're creepy. (Which is something we could've told him a long time ago.) The Time Traveller decides he needs to protect himself, so he starts the long trip to the Palace of Green Porcelain, looking for weapons and a safe haven. On the way to the Palace the Time Traveller realizes that the Morlocks eat the Eloi. It's also on the way to the Palace of Green Porcelain that the Time Traveller spends a sleepless night looking at the stars. While he's doing that, he realizes that human affairs aren't very significant when you take the long (cosmic) view.
From the forest fire to the desolate beach; from the desolate beach back to the smoking room; from the smoking room to…well, where did he go?
In Act III of The Time Machine, the Time Traveller finds the weapons he needs to defend himself against the Morlocks at the Palace. On the way back, he gets the chance to use the weapons. Unfortunately, a fire that he sets that night gets out of control. So now he's running from the monstrous Morlocks and a forest fire. But he survives the night and finds his Time Machine. He goes far into the future, to a nearly lifeless beach, when the earth has stopped turning. Wisely, he decides to go home and tell this story. But his listeners don't really believe him (except the narrator maybe). On his next trip using the Time Machine, the Time Traveller disappears, leaving the narrator to ponder the meaning of the story.