Do I read the Sandman stuff, or do I jump right into the main comics? Because I really want to maintain continuity, what's the right sequence to ensure that I get every bit of the story?
Lucifer starts after the Sandman story.
So if you want to make sure to get the whole story, you'll need to read The Sandman, especially the Season of Mist (The Sandman #21-28). He appears briefly on other occasions, but those are not necessary. (I can't check it right now, but he appears at the end of Preludes & Nocturnes, and in Brief Lives too)
Then you need to read The Sandman presents : Lucifer, which introduces the Lucifer spin off (3 chapters).
And finally, the actual Lucifer comics #1-75.
A one shot chapter exists, Lucifer: Nirvana. It has almost no connection with the rest of the story.
Those 3 series (The Sandman presents: Lucifer, Lucifer, and Nirvana) are all in the correct order, in the collected edition.
I read Sandman some years ago then Lucifer just a couple of months ago, and I think you will enjoy Lucifer more for having read Sandman first.
Reading Sandman gives you Lucifer's back story, and while this doesn't matter that much to the story in Lucifer it does make it more satisfying. I also think Lucifer is deeper than Sandman, and I think Sandman first then Lucifer will be more satisfying than Lucifer first then Sandman (I suppose this doesn't matter if you never plan to read Sandman).
But Sandman is a big publication, and unless you have a cooperative local library it'll cost you a fair bit to buy, then take a fair while to read. If you don't want to invest the time and money in reading Sandman then I'd say you don't have to. There isn't that much in Lucifer that depends on the Sandman story, and Lucifer will still be very enjoyable without it.