Ok, I don't have a ton of time right now so this might suck.
First of all there are a few Batman TPB's that are just universally revered in the comic book community. Moreso than any other character, really. And they're reverted for very good reason. So let's look at a coupe of those absolute classics...
• Batman: Year One - This is an excellent place to start even though it came out 60 years into Batman'a lifespan. A lot of Miller comics become divisive because of that artis's distinct "fuck it, let's go crazy" art style (see Dark Knight Strikes Agaib). Now, while I actually love that style, Year One is much more story-based, and since Miller only writes (leaving art to Dave Mazzucchelli), the look is cleaner and lends itself to the story very well. The overall look is a modern throwback to the 1930s, taking cues from the short black and white films of that era.
It also expands on the early chapter of Batman's career and his life prior to becoming "The Dark Knight," and it gives the reader a nice impression of Gotham as a whole, not just Batman. To that end, we get a lot of Jim Gordon, and his storyline is the backbone of the book. If you ever thought Gordon wasn't interesting, wasn't important, was just a fat guy (Burton movies) or a fat guy with an inexcplicable heavy Irish accent ('60s tv series) this TPB shows you that he's actually one of the best and strongest supporting characters in the Bat-canon. Second only to Alfred, really. You see his set-up as a complex hero-- a man so good and so honest that he will absolutely FUCK SHIT UP if backed against a wall. This take on Gordon is often missing from movies and film (though Gary Oldman nailed it, and the portrayal on Gotham actually does it justice very well), and it sets up a very strong parallel between Gordon and Batman. They are really more alike than either man realizes. Essentially Gordon is the alternate universe Batman, where the army and law enforcement took the place of a mask and ninja training, and where an attempt was made to be a "normal" family man. (Conclusion: maybe Gordon should have just been a vigilante.)
• The Man Who Falls (1989, single-shot, from DC's Secret Orgins) & The Man Who Laughs (2005) - The Man Who Falls is a great little story that should be read alongside Year One. Put them together and you have the plot for Batman Begins. Very enlightening, and quite underrated.
The Man Who Laughs looks back to the Batman's first Joker tangle and mythologizes a bit, painting a really powerful image of the feud between these characters.
• A Death in the Family & The Killing Joke -
So if you're going to dive into any Batman arcs from the past 25 years, you need to have some familiarity with these two titles. They may not necessarily inform everything
thereafter (there are a lot of relaunches and sidestories and different timelines where these events either didn't take place or aren't discussed) these titles certainly weave themselves into a LOT of it.
Both from 1988 (Death... ran from '88-'89 and was a big ass event for DC, maybe you've heard the stories about fans getting to choose Robin's fate?) these comics helped usher in the Batman and Joker we know today. Understanding Joker mythology is almost as important to reading Bat-titles as Batman's own. A lot of people thank The Dark Knight Returns with giving us the darker edgier Batman of the Burton films, and even the Nolan films, but that story was never canon, and it really only lives in the universe of its two sequels. THESE two titles are really what not only pushed Batman into the darkness, but established the Joker as one of the worst and most formidable villains in all of villaindome.
Of course DITF deals with the death of the Jason Todd's Robin (though "death" is of course open to interpretation in comics
). Todd was going down a really fucked up path and was beginning to show signs of mental illness or outright sociopathy. But he didn't deserve what happened to him in this title. Yikes.
Killing Joke is obviously super famous and I'm honestly not sure why sometimes. It goes a little overboard, beyond even the darkest incarnations of the Joker in film, and it was never meant to be canon, but somehow it's been assimilated, and a whole facet of the Bat-world is built on the events in this comic. Anyway, neither of these are favorites of mine, but they are essential reading.
• Arkham Asylum (A Serious House on Serious Earth): one the biggest selling graphic novel of all time, this Grant Morrison/Dave Mckean joint will make you want to read more Batman perhaps more than any other story. Visually, it's stunning. Thematically, it doesn't get much more weird or intense than this. You get to see the world through the Joker's eyes! You follow Batman into a besieged Arkham and you encounter a handful of villains at their most fucking depraved. Its not central to any overarching story, but the TPB will, I believe, give you a lot to look forward to in other titles (sometimes people only go with Joker stories, thinking Hatter, Riddler, etc. will be boring... this clears that up nicely by showing you just how terrifying even lame villains like Killer Croc can be in Batman's world.)
• Batman: Hush - This one is honestly more of a visual pleasure than an intellectual one, but the story gets good and backs up the art in the end.
• Batman: New 52... Just pick up New 52 Batman and read. It's very good.
• Haunted Knight, The Long Halloween & Dark Victory (the Loeb/Sale "trilogy") -
Though Haunted Knigt is really just an anthology piece with no bearing on the story to follow, it introduces Jeph Loeb and Tim Sale as one of the best duos to ever work on Bats. Get through Haunted Knight, and Long Halloween/Dark Victory are virtually as satisfying as any movie. Imagine every Gotham character, in one really long arch, with no cheese. That's how these stories play out. (Follow up with Catwoman: Paris also by Loeb and Sale)
• Court of the Owls - This is one of the best Bat archs of the last several years, plain and simple. The Owls will no doubt be a go-to for future adaptations, as they prove to be an extremely formative force with an iron grip on Gotham. Plus, they date back (in retrospect) to Gotham pre-Batman. Kind of like Bane, the Court is a foe Batman underestimated and was utterly unprepared for.