Edit: I almost forgot, the book is available for free on Gutenberg.org because it is out of copyright in the US (i.e. public domain), so it is free and legal to share with anyone and everyone without limitation. The link to the book is right here!
Chapter 1 is pretty simple. As far as I can tell, this chapter is just providing the literary setting for the story.
Once you strip out the wonderful descriptions, the story is thus: a first-person narrator awakes the day after his 21st birthday, having received all the possessions of his long-dead father (whose death is not described), and he goes to explore some long-unused secretary (a sort of antique desk). As he pokes around, he discovers a miniature-sized living woman and the "Adventures in Wonderland"-esque aspects of the novel abruptly break in. It took me a while to understand what's going on because the narrator takes the experience with a lot more credulity than I had.
How did this woman get here? How had she survived all these years? How is she so small? All of these questions are completely ignored by the story, setting the tenor of much of the rest of the book. This is not a book of logic, facts or "science" (in the modern sense). It is, in fact, fantasies in the traditional sense (wonders, dreams, etc). So the first thing you (my anonymous reader) should do when reading this book is discard any modern set of rules or rationality regarding story development because the author is not behaving very modern (the book is itself about 152 year olds, so it was much less common to be dominated by the modern paradigm at that time).
With that said, if one stays immersed in the descriptions, almost raw sensationalism, of the book, I think the scene is painted wonderfully and proves its worth in that fashion.
We are also given a name for the narrator; Anodos. The woman remains nameless, but after some discussion she increases her size to full human proportions and they continue talking about things past and present. At the end of the discussion (the payload of this whole chapter, really) the woman promises Anodos that he will travel to Fairy Land the next day.
I think the topic of Fairy Land deserves more discussion, because it is a very common trope of 19th and early 20th century literature (and probably earlier as well, but I'm not a literary scholar so I don't really know), but I will forbear on that discussion for now, so that I have enough things to talk about in later chapters.
On to chapter 2!