Most of Tolkien's creative energy was channeled into The Silmarillion and its parts. If you've read and enjoyed The Lord of the Rings, it's only natural that you would want take a peek into the Sil. Unfortunately, most readers bounce off of the book, finding it too difficult to understand. This is an attempt to orient new readers, so they don't put down what might be Tolkien's best writing.
The Silmarillion is a book made up of five separate pieces: Ainulindalë, Valaquenta, Quenta Silmarillion, Akallabêth and Of the Rings of Power and the Third Age. Each piece stands more or less alone, and their styles are very different.
The first section is Ainulindalë. If you've ever heard people claiming that The Silmarillion is like the Bible, this is where they got that impression.
Ainulindalë is the creation myth of Middle-Earth. The style is similar to the first few chapters of Genesis. Here, you'll be introduced to the Valar, and especially to Melkor, who will be major figures going forward. Ainulindalë is short, and the rest of the Sil is not written in this style.
Ainulindalë told you about how the world was created, and how certain spirits who assisted in its creation decided to descend down into it and live there. Valaquenta is a quick run-down of the most powerful among those spirits, the Valar. This section is mostly descriptive, not narrative. Some of these characters only rarely feature in the main narrative, so you may want to reference this section when you forget who's who.
This is the Silmarillion itself. The previous two sections were only introductions to Quenta Silmarillion, and the two following parts are essentially epilogues.
The style of Quenta Silmarillion is similar to an Icelandic Saga, or Herodotus' Histories. This isn't to say that there's anything particularly Norse or Greek about it, but that the way it tells its story is similar. More on this section later.
Quenta Silmarillion described the First Age in fine detail. Akallabêth describes the Second Age in a less detailed manner. The style is more historical than the last, painting the story in broad strokes.
This final section is similar to Akallabêth, being a big-picture historical summary of The Lord of the Rings.
One of the most common difficulties new readers face is the massive number of names that get thrown around. If the sheer number of them weren't bad enough, they all sound similar: Feanor, Finwe, Fingon, Fingolfin, Finrod, Beor, Beorn, Beren, Huor, Tuor, Hurin, Turin, and so on. It can feel overwhelming.
The problem is that The Silmarillion is ultimately not a book about individuals, but about families. This is what makes it similar to a Norse saga. The Lord of the Rings is about specific people, and so keeping track of specific people and their names is important. In fact, that's the case with almost every modern work of fiction. They're about individuals.
In The Silmarillion, though, you will need to keep track of families much more than individuals. This is why characters have similar names. It's to make your reading easier, not harder. Finwe, Fingon, and Fingolfin are hard to keep track of if you don't know who they are. But the similarity of their names tells you what you need: these characters are all related. Likewise for Hurin and Turin, and Huor and Tuor. The names are tricky to distinguish, and that's a good thing. It helps you keep track of who is family.
Why are we watching families instead of individuals? Because in The Silmarillion, the poor choices the father made will inevitably mean something for the children. Daddy Curufinwe makes a rash vow, kills some people he shouldn't have, and it's his son Curufin who's gonna catch the brunt of the karma. Turin's life sucks because Hurin was cursed. This family hates that family, so when the members of the two are together, they fight. The characters are actually the families, less than the individuals.
Three of the chapters of The Silmarillion are concerned with specific individuals who either witness or cause the major turning points in the story. These would be Of Beren and Luthien, Of Turin Turambar and Of Tuor and the Fall of Gondolin. The chapters leading up to these stories are preparing the scene for the action you'll read there. The chapters following them are showing you how the world has changed as a result of these three adventures.
If you ever feel lost while reading The Silmarillion, refer back to these chapters. Are you in the wake of a Great Tale? See how the world is different. Are you coming up to one? Trust that the little pieces are going to come together when you get there.
To put it bluntly, the first time you read this book, you won't get everything. There's simply too much happening to understand it all in one pass. You'll be left after your first read with a sense of awe and an impression of the big picture, and each time you reread, the image will become sharper and sharper. This is how good books are; you can read and reread them with greater pleasure each time.
Once you've read the Sil once, you should talk to other people who have. The most fun part of this book is its fans. Read and get into the discussions, laugh at the memes, enjoy the book with people. It's a lot of fun in itself, but you'll also soak up knowledge passively, and be able to understand much more of the book next time.
That should get you started. Enjoy your reading!