With a cover date of January 1989 (meaning it probably hit comic book stores in November of 1988), Neil Gaiman's Sandman #1 began slowly changing a chunk of the DC Universe into its own corner that would eventually be called DC Vertigo. The Vertigo line would be more literature focused than its mainstream superhero counterpart, but would still share ... not conitnuity ... influence. There are some Justice League members who fall into the Vertigo chasm, but their ambitions seem different when they're in Vertigo titles. And, sure, Death from Sandman, and Swamp Thing and Animal Man cross back and forth between DC Vertigo and DC proper every decade or so, but the two universes were mostly separate. Until. Until Flashpoint and The New 52 in 2011, at which point the Vertigo imprint ceased having its historic value. In January of 2020 Vertigo was completely removed and replaced by DC's "Black Label", which I'm sure they will relaunch again as something else within the next five years.
But the books that inspired the creation of the label: Gaiman's Sandman, Alan Moore's Swamp Thing, and Jamie Delano's Hellblazer are still vital to DC mythology. While books like Y The Last Man, Fables, and Transmetropolitan are important books that came out with the Vertigo label, they would also fit right in with current Image Comics titles or IDW. The early 90s Vertigo books had fun with characters and references crossing between titles, even as the books themselves didn't have Crossover Events. Dream would just show up in Hellblazer to chastise John Constantine. And Constantinem, himself, first showed up in the pages of Swamp Thing. Variations on these characters and these stories went on for almost forty years before they were completely swept into DC proper.
I'm currently reading through as much as I can of the Sandman/Swamp Thing/Hellblazer/Books Of Magic/Dreaming universe in preparation for the upcoming Sandman/Locke & Key crossover.
You shouldn't have to do that. There's a lot of it. Like pretty much every ongoing comic series that lasts more than a couple of years, the quality varies wildly. So I've put together a reading list/order if you want to read Just The Best of these titles. I don't care for which stories are the most Historically Important, like the first appearances of so and so, or The Absolute Beginning of this universe. I care about what's fun to read, and what art is just bonkerstown great.
Most to all of these books should be in print and available from a local comic book store. And if they do dip temporarily out of print, I'm fairly confident they'll return quickly with new cover art and probably a heftier price point. Hope you enjoy!