I'm always posting suggestions in the comments whenever the topic of recommended comic book reading comes up, especially when superheroes are involved. And then I thought, why not just post a list on a semi-regular basis? So here's the first, starting with one of the most popular heroes, Batman. The "get off my lawn!" part is because most of these are going to be older stories.
I'm only going to recommend stories that have been collected into trade paperback form. I am all for supporting local comic shops (sadly, several of the ones from my youth weren't supported enough but I loved going there, and pull lists are recommended). But the sad reality is, not everyone can get to a store, and there's also the digital option for those of us running out of space to store our comics.
These are stories that feature the detective side of Batman, and introduces some characters that would go on to play bigger parts in his stories.
Tales of the Demon
sadly, there don't seem to be any new printings of this, so places like Amazon and Barnes & Noble only sell through 3rd party sellers. A local store might still have copies. It's a collection of Batman stories from the 70s, making Batman more of the Dark Knight Detective we know today, and introducing Talia and Ra's Al Ghul, in stories that will be familiar to fans of The Animated Series, and featuring the League of Assassins. I also highly recommend reading the forward and afterward.
Ten Nights of the Beast
another one that doesn't seem to be in print anymore. This was the first of what would become called a "within series" - a mini series that takes place within a regular series. The forward (recommended reading as well) explains this. The story itself introduces the KGBeast (who would make later appearances in the comics and even in JLU) and features Batman as a detective and master of disguise, and the art of Jim Aparo.
Death in the Family
A turning point for Batman, and DC making use of phone call voting way before American Idol ever aired. (Also available in Kindle Edition)
The Many Deaths of Batman
Takes place just after Death in the Family and gives some background on who helped Bruce Wayne become Batman. One of the issues relies solely on the artwork to convey the story (which it does well) with just 2 lines of dialogue word ballooned. This story is also no longer in print so 3rd party sellers or local stores are the way to go. This is the first story I got when I started collecting/reading comic books as a kid so I have a bias for it.
A Lonely Place of Dying
Introduces Tim Drake, and ran through Batman and Teen Titans titles. Also no longer in print. There is also Batman: Year Three which covers Dick Grayson's start as Robin in the past and Zucko up for parole in the "present". Some of the story seems to have influenced the Robin's Reckoning 2 parter from The Animated Series. But there doesn't seem to be a collection so you'd have to hunt down the issues unless they're available digitally.
Speeding Bullets - an Elseworld's story about what happens when the rocket from Krypton lands in Gotham and Kal-El lives Bruce Wayne's life, including a fateful trip to the theater. It's a great Superman story disguised as a Batman one.
JLA: Tower of Babel
Batman is often referred to as having a plan for everything. This is a great story demonstrating that, by one of my favorite writers, Mark Waid.