In his introduction, Gaiman states that whenever another comic writer would ask him what he thought should spin-off from Sandman, he always told them "Lucifer," which is likely not what any of them wanted to hear, instead hoping for dream assignments (pun intentional) like Death, Desire or Destiny. But, as in most things relating to his comic, Gaiman was spot on. He had already covered the possibilities with the Endless, while Lucifer was the perfect protagonist: extremely flawed yet ultimately intriguing, charming and deadly, full of pride but also hubris. Lucifer is the character you don't want to like, but you can't help yourself from doing so because, unlike the throne of Heaven, Lucifer is much more like us. The Bible may say that man was created in God's image, but our personalities were grafted from Lucifer.
This collection has three main story arcs: "The Morningstar Option," where Lucifer fulfills a job for Heaven, and shows that he's a right bastard to any who doubted; "A Six Card Spread," in which Lucifer attempts to discover if his payment from Heaven has any strings attached that he wasn't aware of; and "Born with the DEAD," which seems ancilliary to the ongoing story, as it's about a young girl who tries to figure out how and why her best friend died with Lucifer showing up near the denounement to provide a favor which will put the girl in debt to him. There's a thread of story that weaves through these three, but, as in Gaiman's Sandman, it will take at least another three or four collections before you start to see the resulting weave that begins here.