What is the Silmarillion, exactly? This is a slightly more complicated question than it seems. The Silmarillion is a book published in 1977, four years after Tolkien's death. The Silmarillion is a collection of stories chronicling the history of the world of which Middle Earth is only a part, written down by Tolkien and edited by Christopher Tolkien and fantasy author Guy Gavriel Kay. The Quenta Silmarillion is the story of First Age and the war of the Silmarils. (0) Confused yet? Of course you are. Don't worry, we're just getting started; you'll be much more confused before we're done.
The Silmarillion is not one story; it's a collection of various things, in different writing styles told with differing levels of detail. There's a sense in which it's reasonable to call it the Bible of Middle Earth: the stories it tells are very old, there are lots of inconsistencies, and no one's sure how much of it to take as Gospel. Tolkien was a tinkerer(1); he was never 100% finished with anything he wrote, even after it was published(2), and the stories that make up The Silmarillion were never finished to his own satisfaction. Fortunately, he also never threw anything away. After his death, when Lord of the Rings was a genuine phenomenon and people were clamoring for more, Tolkien's son Christopher gathered together his father's papers and published the book that Tolkien had started writing decades before but was, honestly, never going to finish.
Many of these stories exist in multiple versions, and it's sometimes hard to tell which version Tolkien actually preferred.(3) He changed names, backgrounds, and genealogies so frequently that some people (okay, some Elves) end up with personal histories that Tolkien almost definitely did not intend, just because we're trying to make sense of the stories we have.(4) You can take this in one of two ways: either you can try to figure out which version is "right" and defend your stance against all comers, or you can treat all of this as part of the glorious mythological history of Middle Earth. No one knows who Gil-galad's father is? That's okay! It's been lost to history, and Elrond's not telling. I like the second strategy.
So you've got a weird mish-mash of things in this book. You've got the creation of the world, which was informed by Tolkien's own Catholicism as well as Nordic myth and his general sense of the aesthetics of the universe. You've got detailed stories like Beren and Luthien or The Children of Húrin which tell important events in the lives of specific people; you've got whole millennia that go by in the space of five pages of migration patterns; you've got genealogies out the wazoo. There are chapters in The Silmarillion that would make lovely animated short film pieces, and there are chapters that would make five-hour epic miniseries plots. We're covering around ten thousand years of history, here.(5)
If you're a completist, you should know that this is by no means all of Tolkien's Legendarium writings. Once you've got the Silmarillion under your belt, your next stop should be Unfinished Tales of Númenor and Middle Earth, which expands on some of the stories in The Silmarillion and offers some other delightful background bits and deleted scenes, like the story where Gandalf tries to make a moral argument out of Thorin Oakenshield being killed by an army of Orcs and Frodo's not having any of it.(6) After that you can dig into the History of Middle Earth wherever seems most interesting to you. Want to see how Tolkien kept track of the phases of the moon throughout The Lord of the Rings? You want Volume 8, The War of the Ring. Love epic poetry? Volume 3, The Lays of Beleriand. And so forth.(7)
So what am I doing here? I am not a Tolkien scholar. I'm not even a particularly qualified Tolkien nerd. But I am the one of my group of friends who's read the most of this auxiliary material, and on the way home from a midnight viewing of The Return of the King, I started explaining the history of Númenor to the rest of the party. They found it entertaining, I found it enjoyable, and here I am, eight years later, still doing it. I hope that you also find it entertaining, and perhaps educational.
(0)Generally people don't italicize it unless they're talking specifically about the book itself, and for the most part "The Silmarillion" will refer to the Quenta Silmarillion stories of the First Age, which is also the largest part of the book. If someone wants to talk about stories in the Akallabêth, which is a section of The Silmarillion, they'll generally use "Akallabêth."