The traditional "right" answer is Frank Miller's Year One, and that's a very good place to start. It's a brilliant introduction for Batman, Gordon, and even Catwoman.
After that, I'd advise you to go through Jeph Loeb's arc that began with Long Halloween, and then encompassed Catwoman: When in Rome, Haunted Knight, and finally Dark Victory. These take place very early in Batman's career, and they also provide a kind of gauntlet of villains. Most of all, though, they tell the story of Harvey Dent's fall from grace--- and I implore you to read them before any other stories that heavily involve Two-Face. You should experience his fall first.
Other than that, I'd also say that while Grant Morrison has written some fantastic work with Batman, I'd recommend leaving most of his stuff until later, until you're very well-versed in Batman and his enemies. Grant Morrison's work goes down better then. The exception to this is Arkham Asylum: Serious House on Serious Earth, which is very dark in tone, but can be read at any time without feeling disjointed.
The Knightfall arc is famous, and rightly so. Along with the story of Azrael and Bane, it also provides an incredibly wide variety of the villains. Much less famous is Tim Sale's Tales of the Batman, and if you're going to read Knightfall, I'd greatly recommend reading Tales afterwards. In part, it tells the story of the Arkham inmates once they're apprehended, but with no Arkham to be sent back to, courtesy of Bane.
Finally, once you've read a few of the more fundamental titles, No Man's Land is also definitely worthy of attention (The lead-ups are Contagion and Cataclysm; I've only read the latter, so don't worry too much). Gotham is a wreck, wracked by disease and earthquake, reduced to rubble and 'excommunicated' from the United States. This is the story that really got me to respect The Penguin as a worthy nemesis.
