1) The Genesis story, that Music stuff I was talking about above. Even as a Tolkien nerd, I find it a bit dry and am not too upset if I skip it
2) The Elves in Valinor and the First Age. This is by far the bulk of the book, and is written kind of as a series of interconnected short stories in mostly chronological order until the end of the First Age.
3) A mini-short story about Numenor.
That's it. For my two cents, the stories told in the main part of the book are far more epic and tragic than anything else in the whole history of the Legendarium.
All that business about the Second Age? That's pieced together from the Numenor part of The Silmarillion, backstory in LOTR, the Appendices in LOTR, and the book of Unfinished Tales. There's no single book or series of books that details all of it, you kind of have to do the work yourself (or else hit up the Wikia), because Tolkien never took the time to assemble it into a single narrative.
Legolas and Gimli hung out together a lot after the War of the Ring, going exploring and sightseeing throughout Middle Earth. As an Elf, the standing invitation to return to Valinor included him, so ultimately he went into the West as well--and brought Gimli with him.
So, two things I forgot. Whoops