Reddit reviews The Books of Magic

Hellblazer. You'll see John Constantine (rhymes with "fine") bumming around a lot of Vertigo stuff. When you read The Sandman, you'll get a bit of him in part one. He's by far my favorite comic book character, and I'm still pissed that the movie didn't do him any justice.

Dangerous Habits, Son of Man, and Setting Sun were some of my favorite collections.

Edit: Forgot to mention The Books of Magic. Yes, he's a dark haired bespectacled lad with an owl who it is foretold will be a powerful wizard, but it predates Harry Potter.

Overall I've found you can't really go wrong with Alan Moore or Neil Gaiman.

• First and foremost, Neil Gaiman's Sandman series. Really, pick up the first volume.. and then I dare you to stop reading.
  
  
• Everything Warren Ellis touches generally turns to gold, but I especially loved his Transmetropolitan series. Science fiction, anarchy, general badassery.. fantastic all around.
  
  
• Ghost World has been mentioned, and while I liked that, I thought David Boring was a superior work from Clowes.
  
  
• Just picked up The Unwritten, a new series by Mike Carey. After finishing the first volume, it looks pretty promising.
  
  
• Demo was dark and angsty, but in a good way.
  
  
• Marvel 1602 was a really interesting take on superheroes.. and again, penned by Gaiman. He fucking rules. Oh, and another Gaiman favourite of mine: Books of Magic.. Gaiman knows the right way to write John Constantine.
  
  Edit: Also, you might want to give manga (asian graphic novels) a try! Good ones to get started with are Death Note and Bleach. I also liked xxxHolic (though I never finished the series), and the anime film for Akira was amazing so I can only imagine that the manga would be just as good -- if not better!

Justice League Dark introduces new versions of old characters. You don't need any prior knowledge to enjoy it, though it's probably better to start at issue 9. The first writer was writing for fans of old Vertigo books, so it's not as great a read for someone with your knowledge.

Neil Gaiman's Books of Magic is a great introduction to all of the DC magic characters, and is Tim Hunter's origin story.
Alan Moore's Swamp Thing Volume IV features Deadman, Phantom Stranger and Constantine.
Both of these are essential reading for DC Dark in my opinion.

However, if you want to know more, I do have recommendations.
Concerning Madame Xanadu, the New 52 Demon Knights does a really good job with her story.

Deadman had a nice run in DC Universe Presents at the beginning of the New 52 (Issues 1-5), but he really shines in his guest appearances in stuff like Books of Magic and Alan Moore's Swamp Thing.

Most of Zatanna's stories take place in Justice League books, but I really enjoyed her in Grant Morrison's Seven Soldiers, which is also Frankenstein's origin story.

For more Frankenstein, I really enjoyed the three issue arc in Flashpoint, despite it not taking place in continuity, and the entire New 52 run.

And of course, if you want more Constantine, you can check out Hellblazer. It's 300 issues long, but you can start wherever a new writer picks up. Just keep in mind that this is a very different character than the New 52 Constantine. This is the version that people have loved over the years, but it won't do much for you if you're researching Trinity War.

However, if you want just enough to get into Trinity War, just read JLD, Phantom Stranger and Constantine from the past few months. None of them actually lead in to Trinity War or anything, but you can know what the characters were up to beforehand.

I think a good introduction is in the first books of magic, and then try Original Sins. If you're not into it then, then it might not be for you. The movie Constantine is okay, but it's not really Hellblazer, and the tv series I hear is more like a procedural. I've vaguely watched John Constantine's appearances in various animated media, they seem like good representations.