Here's the list of the 52 Earths so you can get some idea of which Batman lives where and which Batman stories take place where. I hope this doesn't confuse you too much. I highlighted all the Earths with Batman's that I know of currently:
*
New Earth: Home of Grant Morrison's Batman with Damian (this Batman's son) and Bat-Mite. The Earth that many of the monthly titles today take place on. Created in Infinite Crisis #6
* Earth-2: Home of the Justice Society of America. Revealed in 52 #52
* Earth-3: Home of the Crime Society of America. Revealed in 52 #52
* Earth-4: Home of the former Charlton Heroes/The Watchmen. Revealed in 52 #52
* Earth-5: Home of the Marvel Family. Revealed in 52 #52
* Earth-6: Unknown. Revealed in COUNTDOWN: Arena #2
* Earth-7: Unknown Revealed in COUNTDOWN: Arena #2
* Earth-8: Home of the Extremists. Revealed in Countdown #29
* Earth-9: Home of the Tangent characters. Revealed in Countdown #41
* Earth-10: Home of the Freedom Fighters. Revealed in 52 #52
* Earth-11: Home of Superwoman, Batwoman and Wonder Man. Appearing in Countdown Presents The Search For Ray Palmer: Superwoman/Batwoman #1
*
Earth-12: Home of Batman Beyond and, presumably, DC's animated characters. Revealed in Countdown to Final Crisis #21
* Earth-13: Home of some the characters in the Vertigo titles. Revealed in COUNTDOWN: Arena #1
* Earth-14: ?
* Earth-15: An Earth where society has become nearly perfect. Revealed in Countdown #30
* Earth-16: Home of the Super-Sons. Revealed in COUNTDOWN: Arena #1
* Earth-17: Home of the Atomic Knights and Kamandi. Revealed in 52 #52
* Earth-18: Home of the Justice Riders. Revealed in COUNTDOWN: Arena #2
*
Earth-19: Home of the Victorian era Batman from Gotham By Gaslight by Mike Mignola. Revealed in Countdown #40
* Earth-20: ?
*
Earth-21: Home of the Darwyn Cooke New Frontier characters including the New Frontier Batman. Revealed in COUNTDOWN: Arena #1
*
Earth-22: Home of the Alex Ross Kingdom Come characters including the Kingdom Come Batman. Revealed in 52 #52
* Earth-23: ?
* Earth-24: ?
* Earth-25: ?
* Earth-26: Home of Captain Carrot and the Zoo Crew. Revealed in Captain Carrot & His Amazing Zoo Crew: The Final Ark #1
* Earth-27: ?
* Earth-28: ?
* Earth-29: ?
* Earth-30: Home of the Russian Red Sun Superman. Revealed in Countdown #40
*
Earth-31: Home of the "goddamn Batman" and all the characters in Frank Miller's Batman Year One/All-Star Batman/Dark Knight Returns/DK2. Revealed in COUNTDOWN: Arena #1
* Earth-32: Home of Bruce Wayne/Green Lantern. Revealed in COUNTDOWN: Arena #2
* Earth-33: The Conjurer's World. Home of magical versions of DC characters, including Bat-Mage, Kal-El & Lady Flash. Revealed in Countdown to Adventure #3
* Earth-34: Home of a 19th Century Wonder Woman from Wonder Woman: Amazonia and a Britain run by Jack The Ripper. Revealed in COUNTDOWN: Arena #2
* Earth-35: ?
* Earth-36: ?
* Earth-37: Home of the Thrillkiller characters. Revealed in COUNTDOWN: Arena #2
* Earth-38: Unknown. Revealed in COUNTDOWN: Arena #2
* Earth-39: Unknown. Revealed in COUNTDOWN: Arena #2
* Earth-40: Home of WWII Covert Agents, The Unholy Three. Revealed in COUNTDOWN: Arena #1
* Earth-41: ?
* Earth-42: ?
*
Earth-43: Home of the Vampire Batman from Red Rain, Bloodstorm and Crimson Mist by Doug Moench and Kelley Jones. Revealed in Countdown #40
* Earth-44: ?
* Earth-45: ?
* Earth-46: ?
* Earth-47: ?
* Earth-48: Home of the Forerunners. Revealed in Countdown #46
* Earth-49: ?
* Earth-50: Home of the Wildstorm characters. Revealed in 52 #52
* Earth-51: An Earth where it appears that crime has been eradicated and all super-heroes have retired. Revealed in Countdown #19.
* Earth-52: ?
The new Multiverse has only recently began to be explored. Some of the Earths inhabitance are still unknown - as it is still unknown which Earths are the homes of many other DC characters from the various former Elseworlds series.
I hope that was helpful to your introduction to the DC Multiverse. It's a big place.