If I’m a fan of the tv show, which Lucifer comic runs should I read first?
Well, the show from what I’ve seen is a rather loose adaptation of the character in the comics. He was first introduced in Neil Gaiman’s Sandman, but got his own series by Mike Carey. Sadly, since they’re Vertigo and not connected to any of the DCU shows, it’s not on this service, but it is on Comixology and in trades.
I would recommend checking out The Sandman series where he stars in several stories, and I believe he has a ongoing.
I would start with The Sandman #4 (the Neil Gaiman version) as well as #21-28, the Season in the Mists storyline.
From there the 3 issue The Sandman Presents Lucifer mini-series and then 75 issue Lucifer ongoing series.
It may not be quite the same as the show, but the spirit is there.