The novel starts out with a bunch of guys hanging out after dinner, talking about the nature of time (which is the kind of thing we often talk about after dinner – we're cool like that). They're in Richmond, Surrey (that's in England).
At this get-together are the narrator, the Time Traveller, the Psychologist, the Medical Man, the Provincial Mayor, the Very Young Man, and Filby.
They're chilling – drinking and sitting in ultra-comfy chairs – while the Time Traveller tells them that they don't really understand time.
(The Time Traveller also designed those super ultra-comfy chairs. He's obviously very talented, and we're a little jealous.)
The Time Traveller explains that time is a dimension, just like the three dimensions of space: length, width, and height. The Time Traveller argues that since time is a dimension, we should be able to move along it, into the past or the future.
The others argue with him, but if they were right, this would be a very short book.
The Time Traveller says it's hard to move up or down without any help. Like, you can jump, but that doesn't get you very far up. But he notes that modern people can move up or down more easily because of technology, like hot-air balloons. The Time Traveller concludes that maybe someone will invent a machine to...wait for it...travel through time!
Everyone talks about what they would do if they could travel in time. You could explore great historical events! (And maybe screw them up.) You could go back and invest money in ventures you knew would succeed and become the richest man in the world! (This is an idea Wells played with in another novel, When the Sleeper Wakes.) But the future might not have private property. You could study Ancient Greek in Ancient Greece so you could pass your tests! But since you'd be the only modern person who really knew Ancient Greek, your teachers would fail you. (That's what "plough you for the Little-go" means – it's not as dirty as it sounds.)
Since they don't really believe him, the Time Traveller goes to get his model of the Time Machine from his laboratory.
Filby tries to tell a story, but no one cares.
The Time Traveller shows them his model Time Machine, then makes it disappear.
But some of the dinner guests still don't believe him, so he shows them the full-size Time Machine in his laboratory.
They still don't totally believe him. This is fair, since he's played tricks on them before, like showing them a fake ghost last Christmas. (Today we associate ghost stories with Halloween, but in Wells's day ghost stories were a part of Christmas in Britain.)